1 C H A SME 4463 SME 4463 A P T E R 2 LECTURER: DR. MAZLAN ABDUL WAHID http://www.fkm.utm.my/~mazlan Mazlan 2006 FACULTY OF MECHANICAL ENGINEERING UNIVERSITI TEKNOLOGI MALAYSIA SKUDAI, JOHOR, MALAYSIA HEAT TRANSFER DR MAZLAN C H A Chapter hapter 2 Fundamental Concepts Fundamental Concepts A P T E R 2 Fundamental Concepts Fundamental Concepts of Conduction of Conduction Assoc. Prof. Dr. Mazlan Abdul Wahid Mazlan 2006 FACULTY OF MECHANICAL ENGINEERING UNIVERSITI TEKNOLOGI MALAYSIA SKUDAI, JOHOR, MALAYSIA HEAT TRANSFER DR MAZLAN Assoc. Prof. Dr. Mazlan Abdul Wahid UTM Faculty of Mechanical Engineering Universiti Teknologi Malaysia www.fkm.utm.my/~mazlan
Transcript
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SME 4463SME 4463AAAAAAAA
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LECTURER: DR. MAZLAN ABDUL WAHIDhttp://www.fkm.utm.my/~mazlan
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
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CChapter hapter 22Fundamental Concepts Fundamental Concepts
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Fundamental Concepts Fundamental Concepts of Conductionof Conduction
Assoc. Prof. Dr. Mazlan Abdul Wahid
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
Assoc. Prof. Dr. Mazlan Abdul WahidUTM Faculty of Mechanical Engineering
Universiti Teknologi Malaysiawww.fkm.utm.my/~mazlan
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Conduction Heat Transfer
In this chapter we will learn:
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�The definition of important transport properties an d what governs thermal conductivity in solids, liquid s and gases
�The general formulation of Fourier’s law, applicabl e to any geometry and multiple dimensions
�How to obtain temperature distributions by using
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�How to obtain temperature distributions by using the heat diffusion equation.
�How to apply boundary and initial conditions
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Conduction is simply:
Transfer of energy from more energetic to less energeticparticles of a substance due to interactions betweenparticles. Conduction refers to the transport of energy in a AAAAAAAA
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particles. Conduction refers to the transport of energy in amedium (solid, liquid or gas) due to a temperaturegradient. The physical mechanism is random atomic ormolecular activity
Governed by Fourier’s law.
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22222222• q: heat transfer rate• A: cross -sectional area
L
TkAq
L
cond
cond
∆−=
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• A: cross -sectional area• L: length• k: thermal conductivity• ∆T: temperature difference across conductor
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Thermal Properties of Matter
• Recall from Chapter 1, equation for heat conduction :
Tk
TTkq
∆=−= 21"AAAAAAAA
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L
Tk
L
TTkqx
∆=−= 21"
� The proportionality constant k is a transport property , known as thermal conductivity (units W/m.K)
• Usually assumed to be isotropic (independent of the direction of transfer): k x=ky=kz=k
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Is thermal conductivity different between gases, liquids and solids?
Thermal Conductivity (k) provides an indication of the rate at which rate at which energy is transferred by the diffusion processenergy is transferred by the diffusion process
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Thermal Conductivity - k
• The thermal conductivity of a material is a measure of the ability of the material to AAAAAAAA
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measure of the ability of the material to conduct heat.
• High value for thermal conductivity good heat conductor
• Low value
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• Low value poor heat conductor or
insulator.
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Thermal Conductivities of Materials
• The thermal conductivities of gases such as air vary by a factor of 104 from AAAAAAAA
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factor of 104 from those of pure metalssuch as copper.
• Pure crystals andmetals have the highest thermal conductivities, and
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conductivities, and gases and insulating materials the lowest.
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Thermal Conductivities and Temperature
• The thermal conductivities of materials vary with temperature. AAAAAAAA
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temperature.• The temperature
dependence of thermal conductivity causes considerable complexity in conduction analysis.
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conduction analysis.• A material is
normally assumed to be isotropic.
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Thermal Conductivity: Fluids
• Physical mechanisms controlling thermal conductivit y not well
understood in the liquid state
• Generally k decreases with increasing temperature ( exceptions AAAAAAAA
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• Generally k decreases with increasing temperature ( exceptions
glycerine and water)
• k decreases with increasing molecular weight.
• Values tabulated as function of temperature.
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Thermal Conductivity: Insulators
– Can disperse solid material throughout an air space – fiber,
How can we design a solid material with low thermal conductivity?
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– Can disperse solid material throughout an air space – fiber, powder and flake type insulations
– Cellular insulation – Foamed systems� Several modes of heat transfer involved (conduction , convection,
radiation)� Effective thermal conductivity: depends on the ther mal
conductivity and radiative properties of solid mate rial, volumetric fraction of the air space, structure/morphology (op en vs. closed
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fraction of the air space, structure/morphology (op en vs. closed pores, pore volume, pore size etc.) Bulk density (s olid mass/total volume) depends strongly on the manner in which the solid material is interconnected.
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Thermal DiffusivityThermophysical properties of matter:
• Transport properties: k (thermal conductivity/heat transfer), νννν(kinematic viscosity/momentum transfer), D (diffusi on coefficient/mass transfer) AAAAAAAA
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coefficient/mass transfer)• Thermodynamic properties, relating to equilibrium s tate of a
system, such as density, ρρρρ and specific heat c p.– the volumetric heat capacity ρρρρ cp (J/m3.K) measures the
ability of a material to store thermal energy.• Thermal diffusivity αααα is the ratio of the thermal conductivity to the
heat capacity:
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pck
ρρρραααα ====
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Thermal Diffusivity
2Heat conducted (m s)
Heat stored
k
cα
ρ= =
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• The thermal diffusivity represents how fast heat diffuses through a material.
• Appears in the transient heat conduction analysis.• A material that has a high thermal conductivity or a low
(m s)Heat stored pc
αρ
= =
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• A material that has a high thermal conductivity or a low heat capacity will have a large thermal diffusivity.
• The larger the thermal diffusivity, the faster the propagation of heat into the medium.
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The Conduction Rate Equation
Recall from Chapter 1:
dx
dTkAQx −=• Heat rate in the
x-direction qxAAAAAAAA
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dx
• Heat flux in the x-direction dx
dTk
A
qqx −==
We assumed that T varies only in the x-direction, T=T(x)T1(high) qx” A
qx”
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Direction of heat flux is normal to cross sectional area A, where A is isothermal surface (plane normal to x-direction)
T2 (low)
x1 x2
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The Conduction Rate EquationIn reality we must account for heat transfer in three dimensions• Temperature is a scalar field T(x,y,z)• Heat flux is a vector quantity. In Cartesian coordinates:
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zyxxqqq kjiq ++++++++====
for isotropic mediumzT
kqyT
kqxT
kq zyx ∂∂∂∂∂∂∂∂−−−−====
∂∂∂∂∂∂∂∂−−−−====
∂∂∂∂∂∂∂∂−−−−==== , ,
TkTTT
k ∇−=
∂∂+
∂∂+
∂∂−=∴ kjiq
” ”” ”
”
”
” ”
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Tkzyx
k ∇−=
∂
+∂
+∂
−=∴ kjiq
Where three dimensional del operator in cartesian coordinates:
zyx ∂∂+
∂∂+
∂∂=∇ kji
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Summary: Fourier’s Law
• It is phenomenological, ie. based on experimental evidence
• Is a vector expression indicating that the heat flux is normal to an AAAAAAAA
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isotherm, in the direction of decreasing temperature
• Applies to all states of matter
• Defines the thermal conductivity, ie.
)/( xTq
k x
∂∂∂∂∂∂∂∂−−−−≡≡≡≡
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)/( xT ∂∂∂∂∂∂∂∂
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The Heat Diffusion Equation
• Objective to determine the temperature field, ie. temperature distribution within the medium.
• Based on knowledge of temperature distribution we can compute the AAAAAAAA
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• Based on knowledge of temperature distribution we can compute the conduction heat flux.
� Reminder from fluid mechanics: Differential control volume.
We will apply the energy conservation equation to
Element of volume:dx dy dz
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conservation equation to the differential control volumeCV
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
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zq
yq
dzzqdyyqdxxqoutE
zqyqxqinE
+++++=
++=
&
&
zz ∂∂x
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Heat Diffusion Equation
• Thermal energy generation due to an energy source :– Manifestation of energy conversion process (between
thermal energy and chemical/electrical/nuclear energy) ) (
V
dzdydxq
qEg
•
••
=
=
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energy)� Positive (source) if thermal energy is generated� Negative (sink) if thermal energy is consumed
) ( dzdydxq=
• Energy storage term– Represents the rate of change of
) ( dzdydxtT
cE pst
∂∂∂∂∂∂∂∂==== ρρρρ&
q•
is the rate at which energy is generated per unit volume of the medium (W/m 3)
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– Represents the rate of change of thermal energy stored in the matter in the absence of phase change.
t ∂∂∂∂
tTc p ∂∂∂∂∂∂∂∂ /ρρρρis the time rate of change of the sensible (thermal) energy of the medium per unit volume (W/m 3)
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Heat Diffusion Equation
Substituting into Eq. (1.11c):
t
Tcq
z
Tk
yy
Tk
yx
Tk
x p ∂∂=+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂ •
ρ
Heat Equation
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tcq
zk
yyk
yxk
x p ∂=+
∂∂
+
∂∂
+
∂∂
ρ
Net conduction of heat into the CVrate of energy generation per unit volume
time rate of change of thermal energy per unit volume
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unit volume
�At any point in the medium the rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume
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Heat Diffusion Equation- Other forms
• If k=constant
t
T
k
q
z
T
y
T
x
T
∂∂=+
∂∂+
∂∂+
∂∂
•
α1
2
2
2
2
2
2is the thermal diffusivity
pc
k
ρ=α
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tkzyx ∂=+
∂+
∂+
∂ α222p
• For steady state conditions
0=+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂ •
qz
Tk
yy
Tk
yx
Tk
x• For steady state conditions, one-dimensional transfer in x-direction
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• For steady state conditions, one-dimensional transfer in x-direction and no energy generation
0or 0 ==
dx
dq
dx
dTk
dx
d x � Heat flux is constant in the direction of transfer
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t
T
k
qT
∂∂=+∇
•
α12
Fourier – Biot Equation
T∂1
Heat Equation- Other forms
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t
TT
∂∂=∇
α12
02 =+∇•
k
qT
Heat Diffusion Equation
Poison Equation
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k
02 =∇ T Laplace Equation
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2
2
2
2
2
22
z
T
y
T
x
TT
∂∂+
∂∂+
∂∂=∇
In rectangular coordinate:
In different coordinateT2∇
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zyx ∂∂∂
2
2
2
2
22
22 11
z
TT
rr
T
rr
TT
∂∂+
∂∂+
∂∂+
∂∂=∇
φ
In cylindrical coordinate:
In spherical coordinate:
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∂∂
∂∂+
∂∂+
∂∂
∂∂=∇
θθ
θθφθT
r
T
rr
Tr
rrT sin
sin
1
sin
1122
2
222
22
In spherical coordinate:
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Heat Diffusion Equation
• In cylindrical coordinates:
Tcq
Tk
Tk
Tkr p ∂
∂=+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂ •
ρφφ
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tcq
zk
zk
rrkr
rr p ∂=+
∂∂
+
∂∂
+
∂∂
ρφφ2
• In spherical coordinates:
t
Tcq
Tk
r
Tk
rr
Tkr
rr p ∂∂=+
∂∂
∂∂+
∂∂
∂∂+
∂∂
∂∂ •
ρθ
θθθφφθ
sinsin
1
sin
11222
22
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trrrrr ∂ ∂∂
∂∂ ∂∂ θθθφφθ sinsin 2222
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Cylindrical Coordinates
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••= qegen
with
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2
1 1gen
T T T T Trk k k e c
r r r r z z tρ
φ φ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂
&
(2-43)
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22222222••
= qg
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= qg
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Spherical Coordinates
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••= qegen
with
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22 2 2 2
1 1 1sin
sin sin gen
T T T Tkr k k e c
r r r r r tθ ρ
θ φ φ θ θ θ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + + = ∂ ∂ ∂ ∂ ∂ ∂ ∂
&
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••= qg
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Boundary and Initial Conditions
• Heat equation is a differential equation:– Second order in spatial coordinates: Need 2 boundary conditions– First order in time: Need 1 initial condition
Boundary Conditions AAAAAAAA
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Boundary Conditions
1)1) FIRST KIND (DIRICHLET CONDITION):FIRST KIND (DIRICHLET CONDITION):
Prescribed temperaturePrescribed temperatureExample: a surface is in contact with a melting solid or a boiling liquid Ts
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boiling liquid
x
T(x,t)
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Specified Temperature Boundary Condition
For one-dimensional heat transfer through a plane wall of thickness AAAAAAAA
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through a plane wall of thickness L, for example, the specified temperature boundary conditions can be expressed as
T(0, t) = T1
T(L, t) = T
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1
T(L, t) = T2
The specified temperatures can be constant, which is the case for steady heat conduction, or may vary with time.
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Boundary and Initial Conditions
2) SECOND KIND (NEUMANN CONDITION): 2) SECOND KIND (NEUMANN CONDITION): Constant heat flux at the surfaceConstant heat flux at the surface
Example: What happens when an electric heater is AAAAAAAA
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attached to a surface? What if the surface is perfectly insulated?
xT(x,t)
qx”
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Specified Heat Flux Boundary Condition
The heat flux in the positive x-direction anywhere in the medium,
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q kdx
= − =&Heat flux in the positive x-direction
including the boundaries, can be expressed by Fourier’s law of heat conduction as
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x-directionThe sign of the specified heat flux is determined by inspection: positive if the heat flux is in the positive direction of the coordinate axis, and negative if it is in the opposite direction.
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Two Special Cases
Insulated boundary Thermal symmetryAAAAAAAA
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(0, ) (0, )0 or 0
T t T tk
x x
∂ ∂= =∂ ∂
( ),2 0LT t
x
∂=
∂
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Boundary and Initial Conditions
3) THIRD KIND (MIXED BOUNDARY CONDITION) : 3) THIRD KIND (MIXED BOUNDARY CONDITION) : When When convectiveconvective heat transfer occurs at the surfaceheat transfer occurs at the surface
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T(x,t)
T(0,t)hT ,∞
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FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
T(x,t)x
CCCCCCCCHHHHHHHH
AAAAAAAA
Interface Boundary Conditions
At the interface the requirements are:(1) two bodies in contact must have the same temperatureat
the area of contact, AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
the area of contact,(2) an interface (which is a
surface) cannot store any energy, and thus the heat fluxon the two sides of an interface must be the same.
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN0 0( , ) ( , )A B
A B
T x t T x tk k
x x
∂ ∂− = −∂ ∂
TA(x0, t) = TB(x0, t)
and
21
CCCCCCCCHHHHHHHH
AAAAAAAA
Generalized Boundary Conditions
In general a surface may involve convection, radiation, and specified heat flux simultaneously. The boundary condition in such cases is again obtained from a surface energy balance, expressed as AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
expressed as
Heat transferto the surfacein all modes
Heat transferfrom the surface
In all modes
=
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
The quantities of major interest in a medium with heat generation are the surface temperature Ts and the maximum temperature Tmax
that occurs in the medium in steady operation.
CCCCCCCCHHHHHHHH
AAAAAAAA
Variable Thermal Conductivity, k(T)
• The thermal conductivity of a material, in general, varies with
AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
material, in general, varies with temperature.
• An average value for the thermal conductivity is commonly used when the variation is mild.
• This is also common practice
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
• This is also common practice for other temperature-dependent properties such as the density and specific heat.
22
CCCCCCCCHHHHHHHH
AAAAAAAA
Variable Thermal Conductivity for One-Dimensional Cases
When the variation of thermal conductivity with temperature k(T) is known, the average value of the thermal conductivity in
AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
2
1
2 1
( )T
Tave
k T dTk
T T=
−∫
the temperature range between T1 and T2 can be determined from
The variation in thermal conductivity of a material with can often be approximated as a linear function and expressed as
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
often be approximated as a linear function and expressed as
0( ) (1 )k T k Tβ= +ββββ the temperature coefficient of thermal conductivity.
CCCCCCCCHHHHHHHH
AAAAAAAA
Variable Thermal Conductivity
• For a plane wall the temperature varies linearly
AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
temperature varies linearlyduring steady one-dimensional heat conduction when the thermal conductivity is constant.
• This is no longer the case when the thermal conductivity changes with temperature
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
changes with temperature (even linearly).
23
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
CCCCCCCCHHHHHHHH
AAAAAAAA
Steady versus Transient Heat Transfer
• Steady implies no change with time at any point within the medium
AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
the medium
• Transient implies variation with time or time dependence
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
24
CCCCCCCCHHHHHHHH
AAAAAAAA
Multidimensional Heat Transfer
• Heat transfer problems are also classified as being:– one-dimensional, AAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
– one-dimensional,– two dimensional,– three-dimensional.
• In the most general case, heat transfer through a medium is three-dimensional . However, some problems can be classified as two- or one-
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
problems can be classified as two- or one-dimensional depending on the relative magnitudes of heat transfer rates in different directions and the level of accuracy desired.
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
25
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
26
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
27
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
28
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
DR MAZLAN
CCCCCCCCHHHHHHHH
AAAAAAAAAAAAAAAA
PPPPPPPP
TTTTTTTT
EEEEEEEE
RRRRRRRR
22222222
Mazlan 2006
FACULTY OF MECHANICAL ENGINEERINGUNIVERSITI TEKNOLOGI MALAYSIASKUDAI, JOHOR, MALAYSIA HEAT TRANSFER
