ME 504 – Advanced Heat Transfer I
Chapter 1 – Introduction and Review
of Fundamentals
Dr. Merve Erdal
Mechanical Engineering Department
Middle East Technical University
(e-mail: merdal@metu.edu.tr, off: ME B-304, ph: 210-5237)
1
Heat
Heat: form of energy that crosses the boundary of a system by virtue
of a temperature difference existing between the system and its
surroundings.
Heat: energy in transition across the system boundary, driven by
temperature difference for its propagation.
Heat flow: vectorial from higher toward lower temperatures
Heat transfer: branch of engineering science that studies transfer of
energy solely as a result of temperature differences.
2
Heat Transfer vs. Thermodynamics
Laws of thermodynamics
- used to predict the gross amount of heat transferred to or from a system
during a process in which the system goes from one thermodynamic state (i.e.,
mechanical and chemical, as well as thermal equilibrium) to another.
Thermodynamics alone, not sufficient
- to provide the time period over which the transfer of heat occurs or the time
rate at which it takes place
- to explain the mechanisms of heat transfer, which are not strictly restricted
to equilibrium states
Science of heat transfer
- studies mechanisms of heat transfer
- calculates heat transfer rates by extending thermodynamic analysis, through
the development of necessary empirical and analytical relations
- enables the determination of temperature
3
Science of Heat Transfer
- based on both theory and experiment.
- theoretical part constructed from one or more physical (or natural)
laws.
Physical laws: statements, in terms of various concepts, found to be
valid through many years of experimental observations.
general law
(its application independent of the nature
of the medium under consideration)
particular law
(its application dependent on the nature
of the medium under consideration)
Almost all heat transfer analyses,
Particular laws employed in analysis of
directly or indirectly, depend on the
conduction heat transfer:
below general laws:
1. Fourier’s law of heat conduction
1. Conservation of mass
2. Newton’s law of cooling
2. Newton’s second law of motion
3. Stefan–Boltzmann’s law of radiation
sufficient for
3. First law of thermodynamics
4. Second law of thermodynamics most heat
conduction
problems
4
Modes of Heat Transfer
Two basic and distinct modes of heat transfer (mechanisms by which heat is
transferred):
1. (Heat) Conduction
- heat transfer by molecular motion, sometimes supplemented by flow of
free electrons and lattice vibrations, through a body (solid, liquid, or
gas) from regions of high temperature to regions of low temperature
- also takes place across the interface between two material bodies in
contact when they are at different temperatures.
2. (Thermal) Radiation
- transfer of heat in the form of electromagnetic waves.
- all substances emit radiation as a result of their temperature
- all substances also capable of absorbing radiation
- radiation can pass through transparent and semitransparent materials
and through vacuum (for heat conduction, a material medium is
necessary)
5
Modes of Heat Transfer (Cont.’d)
• If macroscopic fluid motion exists, internal energy can be transported by
the movement of the fluid itself (advection).
• The process of energy transport by the combined effect of heat conduction
(and radiation), and the macroscopic movement of fluid is referred to as
convection or convective heat transfer.
Convection: not a different mechanism of heat transfer but due to its complex
nature, requires more involved analysis that includes simultaneous study of
fluid motion with the energy transport. Commonly referred to as another
mode of heat transfer
• Temperature distribution controlled by the combined effect of all modes of heat
transfer; not possible to entirely isolate one mode from interactions with other
modes.
• For simplicity, these modes studied separately during analysis.
• Focus of this course: The study of conduction heat transfer
6
Continuum Approach
• Applying physical laws to individual particles (molecules, atoms and
electrons of matter) or a statistical group of particles à gives insight to
details of heat conduction but too cumbersome for most engineering
problems
• Instead, if
• size and mean free path of molecules small compared to other
medium dimensions (statistical averaging is meaningful), i.e.
𝜆 (𝑚𝑒𝑎𝑛 𝑓𝑟𝑒𝑒 𝑝𝑎𝑡ℎ 𝑜𝑓 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒𝑠)
Knudsen number, 𝐾𝑛 =
≪1
𝐿 (𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟𝑖𝑠𝑡𝑖𝑐 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑚𝑒𝑑𝑖𝑢𝑚)
• primary interest not in molecular behavior of a substance but rather
in how the substance behaves as a continuous medium
then, (conduction) medium can be considered to be a continuous
medium, or continuum.
7
Continuum Approach (Cont.’d)
• In continuum approach,
•
(conduction) medium fits the definition of a field where
properties can be represented by continuous mathematical
functions,
e.g. Temperature represented as T(x,y,z,t) has a value at all points (any
(x,y,z) coordinate) at each time instant t – there are no undefined
temperature locations in the (continuous) medium
•
In this course, we will base our analysis on continuum assumption.
8
System vs. Control Volume
System arbitrary collection of matter of fixed identity bound by a (real or
imaginary) closed surface.
surroundings: anything else interacting with the system in consideration
General laws are always stated for a system and are meaningful only when
applied to a system; ”Reynolds transport theorem” allows expressing of
general laws (written for a system) for a control volume.
Control volume
• a specific region (volume) in space bound by a (real or imaginary)
surface (the control surface); may be fixed, moving or deforming
• matter, energy, momentum may flow across control surface
control volume
(fixed in this case)
gas inside the cannister
at time t1 defined as the
system in consideration.
mass in control
volume less at
time, t2
system at later time, t2
(its shape, volume and
position changed but
its mass has not)
9
Further Definitions
Thermodynamic state described by a list of the values of all system properties.
Equilibrium state when all system properties remain unchanged
Process is change of state, described in part by the series of states passed through
by system.
Thermal equilibrium when no energy transfer as heat takes place between two
systems in contact; the two systems have the same temperature if they are in
thermal equilibrium with each other
Work when an energy form of one system (such as kinetic, potential, or internal
energy) is transformed into an energy form of another system or surroundings
without the transfer of mass from the system and not by means of a temperature
difference
Specific heat at constant volume is change in internal energy of unit mass, per
degree change of temperature between two equilibrium states of the same volume
Specific heat at constant pressure is change in enthalpy of unit mass between two
equilibrium states at the same pressure, per degree change of temperature
10
Specific Heat
e.g. How much energy must be added to raise the temperature of 5 kg of
water from 20°C to 100°C at atmospheric conditions ? Assume constant
specific heat at 4.2 kJ/kgK.
𝐸 = 𝑚 𝑐! Δ𝑇 = 5 𝑘𝑔 4.2 𝑘𝐽/𝑘𝑔𝐾 100°𝐶 − 20°𝐶 = 1680 𝑘𝐽
Note that specific heat is temperature-dependent. However, in the given
range of temperature for water, its value does not change significantly
𝑐! (𝑘𝐽/𝑘𝑔𝐾)
15
4.18
4.22
20
100
360
𝑇 (°𝐶)
11
Statement of General Laws
Decide if the analysis will be performed for a system or a control volume
(through Reynolds Transport theorem).
A general law may be formulated for
• lumped systems (if not space dependent, e.g. 𝑇(𝑡) ), or
• distributed systems (if space dependent, e.g. 𝑇(𝑥, 𝑦, 𝑡) or 𝑝(𝑟, 𝜃, 𝑧))
In distributed systems, most commonly used general law statements:
•
integral formulation
•
differential formulation
12
Conservation of Mass
For a system mass m,
Develop the form of this law as it applies to a control volume:
system at time t+Dt
system at time t (same
volume as CV)
2
𝑚+,+
System mass remains constant
(Conservation of Mass):
𝑡
=
𝑚+,+ 𝑡 + Δ𝑡
-.!" /
-.!" /01/ 0.# /01/ 2.$ /01/
𝑚%& 𝑡 + Δ𝑡 − 𝑚%& 𝑡
𝑚' 𝑡 + Δ𝑡
𝑚* 𝑡 + Δ𝑡
= lim
− lim
!"→$
!"→$
!"→$
Δ𝑡
Δ𝑡
Δ𝑡
lim
1
CV
)̇
entering %&
𝜕𝑚AB
+ 𝑚̇ CD/ − 𝑚̇ EF = 0
𝜕𝑡
)̇
leaving %&
Conservation of Mass
expressed for a CV
13
Conservation of Mass (Cont.’d)
𝜕𝑚AB
𝜕𝑡
G outward from CV
unit surface normal 𝐧
G>0
• If outflow, 𝐕 J 𝐧
G<0
• If inflow, 𝐕 J 𝐧
𝐕
!
𝐧
Conservation of Mass
in integral form
time rate of increase
of mass in CV
CV
CS
rate of mass
leaving CV
Converting the surface integral into a volume integral via Gauss’ divergence theorem
as
conservation of mass in differential form (the continuity equation) is obtained as
Continuity Equation
14
Reynolds Transport Theorem
Any time rate of change of a system property 𝐵 (Lagrangian description) can be expressed
as the material or substantial derivative of the same property (Eulerian description) in a
control volume that coincides with the system volume instantaneously, through Reynolds
Transport Theorem:
𝑑𝐵+,+ 𝐷𝐵AB 𝜕𝐵AB
=
=
+ 𝐵̇ CD/ − 𝐵̇ EF
𝑑𝑡
𝐷𝑡
𝜕𝑡
If 𝐵 is an extensive property and its corresponding intensive property is 𝑏, then
𝐵 = + 𝑏 𝑑𝑚 = + 𝜌 𝑏 𝑑𝑉
.
B
1 𝑑𝐴
𝐵̇ = + 𝑏 𝑑𝑚̇ = + 𝜌 𝑏 𝐕 0 𝐧
.̇
• If outflow, 𝐵:̇ positive
• If inflow, 𝐵:̇ negative
N
and the time rate of change of property 𝐵 for a CV (i.e. material/substantial
derivative) can be expressed as
𝑑𝐵+,+ 𝐷𝐵AB
𝜕
1 𝑑𝐴
=
=
+ 𝜌 𝑏 𝑑𝑉 + + 𝜌 𝑏 𝐕 0 𝐧
𝑑𝑡
𝐷𝑡
𝜕𝑡
AB
AO
15
Reynolds Transport Theorem (Cont.’d)
time rate of increase
of system property
𝐵 at time instant 𝑡
time rate of increase
of property 𝐵 in CV
at time instant 𝑡
𝑑𝐵+,+ 𝐷𝐵AB
𝜕
1 𝑑𝐴
=
=
+ 𝜌 𝑏 𝑑𝑉 + + 𝜌 𝑏 𝐕 0 𝐧
𝑑𝑡
𝐷𝑡
𝜕𝑡
AB
material (substantial)
derivative of property 𝐵 in CV
that occupies the same volume
as the system at time instant 𝑡
AO
rate of property 𝐵 exiting
the CV across its CS at
time instant 𝑡
Q
• Note that the substantial (material) derivative operator Q/ can be shown to be
𝐷
𝜕
= + (𝐕 0 ∇)
𝐷𝑡 𝜕𝑡
which can apply to any point function (such as an intensive property 𝑏) as:
𝐷𝑏 𝜕𝑏
=
+ (𝐕 0 ∇)𝑏
𝐷𝑡 𝜕𝑡
16
Conservation of Energy (1st Law of Thermo)
• System undergoing a cyclic process (beginning and ending states the same):
à both integrals path dependent
net heat
transferred
to the
system
net work
done by
the
system
• For infinitesimal change of state over a time interval dt :
increase in
the energy
E of the
system
both are rates!
In the absence of other types of energy (e.g. mass–energy conversion, chemical
reactions, etc.),
E = KE (kinetic energy) + PE (potential energy) + U (internal energy)
U : associated with molecular/atomic behavior of system.
17
Conservation of Energy (Cont.’d)
• Define specific energy, 𝑒:
𝐸 = + 𝑒 𝑑𝑚 = + 𝜌 𝑒 𝑑𝑉
Using Reynolds Transport Theorem ,
.
B
𝑑𝐸+,+ 𝐷𝐸AB
𝜕
1 𝑑𝐴
=
=
+ 𝜌 𝑒 𝑑𝑉 + + 𝜌 𝑒 𝐕 0 𝐧
𝑑𝑡
𝐷𝑡
𝜕𝑡
AB
AO
̇ part of 𝑊̇ due to work done by system against pressure (flow work):
• About 𝑊:
𝛿𝑊TUCV =
𝑝 𝑑𝐴
1 𝑑𝑡
𝐕0𝐧
!WX++DWX TCWYX
ZE+/[FYX /W[\XUXZ C\XW Z/
1 𝑑𝐴
→ 𝑊̇ TUCV = 𝑝 𝐕 0 𝐧
̇ 𝑊̇ TUCV + 𝑊̇ WX+/ (𝑊̇ WX+/ : shaft work, work against shear stress, any work done on
à 𝑊=
system that results in heat generation (−𝐸̇]XF ), etc.)
• With the specific properties, 𝑒 = 𝑢
!
ℎ = 𝑢 + S,
B#
+
R
+ 𝑔𝑧 and the definition of specific enthalpy,
𝜕
𝑉R
1 𝑑𝐴 = 𝑞AB − 𝑊̇ WX+/
+ 𝜌 𝑒 𝑑𝑉 + + 𝜌 (ℎ +
+ 𝑔𝑧) 𝐕 0 𝐧
𝜕𝑡
2
AB
AO
Conservation of
Energy for a CV
18
2nd Law of Thermodynamics
Clausius statement: It is impossible for a self-acting system unaided by an external agency
to move heat from one system to another at a higher temperature à leads to recognition
of system property, entropy
• For a process that a system undergoes during a time interval dt, the rate of
change in the entropy S of the system:
(“=“ for reversible process, “>” for irreversible process)
• Control volume form of 2nd law (via Reynolds Transport Theorem):
19
Temperature
We are interested in temperature distribution or temperature field (instantaneous
values of temperature at all points of medium of interest )
• T(x,y,z,t): 3-D, unsteady temperature distribution in Cartesian coordinates
•
T(r,t): 1-D, unsteady temperature distribution in cylindrical/spherical
coordinates
• T(x,z): 2-D, steady temperature distribution in Cartesian coordinates;
𝜕𝑇
𝜕𝑇
= 0 steady
= 0 (not a function of y)
𝜕𝑡
𝜕𝑦
• T(t): Lumped system; homogenous, unsteady temperature distribution;
temperature is uniform everywhere
Isothermal surfaces: surfaces formed by connecting same temperature points in a
medium
Isotherms : lines formed by intersection of isothermal surfaces with a plane
• Two isothermal surfaces or isotherms can never intersect (a point in a
medium cannot have two different temperatures at the same time)
20
Fourier’s Law of Heat Conduction
A solid flat plate of thickness L (L << other two dimensions) à T changes in x
direction only (1-D conduction)
A: surface area of plate
T1 ≠ T2 ; in this configuration, T1 > T2 and T1, T2 both constant (steady conduction)
Experimental observation:
Definition of thermal conductivity of the plate
material, k:
𝑊
[𝑘] =
𝑚𝐾
Across the infinitesimal plate element of thickness, Dx:
Fourier’s law of heat
conduction for a 1-D
steady system
21
Fourier’s Law of Heat Conduction (Cont.’d)
Heat flux, 𝑞^^ :
𝑞^^ =
𝑞
𝑑𝑇
= −𝑘
𝐴
𝑑𝑥
𝑞^^ =
𝑊
𝑚R
For multidimensional heat conduction, 𝑞 and 𝑞^^ are vectors:
𝜕𝑇
𝜕𝑇
𝜕𝑇
^^
^^
^^
^^
𝒒 = −𝑘∇𝑇 = 𝑞_ 𝒊 + 𝑞, 𝒋 + 𝑞` 𝒌 = −𝑘
𝒊−𝑘
𝒋−𝑘
𝒌
𝜕𝑥
𝜕𝑦
𝜕𝑧
∇𝑇: Temperature gradient (also a vector)
• |∇𝑇|: largest change in temperature per distance (length)
• Direction of ∇𝑇: direction along which temperature increases the most
• How to find the rate of temperature change per length along a specific direction?
Using the directional derivative of temperature.
Let 𝒃 be the unit vector denoting the direction along which we want to find the
temperature change per length. Then, temperature change per length along 𝒃:
𝜕𝑇
= ∇𝑇 0 𝒃
𝜕𝑏
Note that the above is not a vector.
22
Fourier’s Law of Heat Conduction (Cont.’d)
1
𝒒^^ = −𝑘∇𝑇 = 𝑞F^^𝒏
1 : unit normal to
𝒏
isotherms in the
direction of
decreasing
temperature
𝑞F^^ = −𝑘
𝜕𝑇
1 = 𝒒^^ 0 𝒏
1
= −𝑘 ∇𝑇 0 𝒏
𝜕𝑛
23
Example
The temperature field in a conduction medium is 𝑇 = 20 − 3𝑥𝑦𝑡 (where 𝑇
is in ℃, 𝑥, 𝑦 are in m, 𝑡 in s) and the thermal conductivity of the medium
is constant at 2 W/mK.
a) What is the unit vector along the direction in which the temperature
decreases the most at point (1,0,2) at time 𝑡 = 2 𝑠?
b) Find the largest temperature decrease per distance and the heat flux at
the same point and time instant.
c) Determine the temperature change per distance along the direction
𝒊 + 2 𝒋 at the same point and time instant. Also determine the heat
flux along this direction.
24
Solution
25
About Temperature Gradient
26
Thermal Conductivity
Anisotropic conduction medium: thermal conductivity may be have different values
in different directions (anisotropic thermal conductivity) – in heterogeneous media
such as composite materials.
Suppose an anisotropic thermal conductivity has the principal directions, x1, x2
and x3 (perpendicular to one another but not necessarily coinciding with a standard
coordinate system) à Then, 𝑘: a tensor
𝑘aa
𝒌 = 0
0
0
𝑘RR
0
0
0
𝑘bb
𝑘aa, 𝑘RR, 𝑘bb: conductivity coefficients along the principal directions (x1, x2 and x3)
Fourier’s Law of Conduction:
𝒒^^
𝜕𝑇
𝜕𝑇
𝜕𝑇
= 𝒌 0 ∇𝑇 = −𝑘aa
𝒆_ −𝑘RR
𝒆_ −𝑘bb
𝒆_
𝜕𝑥a 𝟏
𝜕𝑥R 𝟐
𝜕𝑥b 𝟑
c$%%
c#%%
• heat flux vector 𝒒^^ not necessarily parallel to ∇T !
c&%%
27
Thermal Conductivity (Cont.’d)
If the coordinate axes used in analysis not coinciding with principal directions:
𝑘aa
𝒌 = 0
0
0
𝑘RR
0
0
0
𝑘bb
→
⏟
𝒌 =
WC/[/EF] [_X+
TWC. _$ ,_# ,_&
/C _,,,` (+E.EU[WE/,
/W[F+TCW.[/ECF)
à Fourier’s Law of Conduction:
𝑘__
𝑘,_
𝑘`_
𝑘_,
𝑘,,
𝑘`,
𝑘_`
𝑘,`
𝑘``
[ +,..X/WEY (j'( -j(' , X/Y.)
kD/ FCFZE[]CF[U .[/WE_
c'%%
𝜕𝑇
𝜕𝑇
𝜕𝑇
= 𝒌 0 ∇𝑇 = − 𝑘__
+ 𝑘_,
+ +𝑘_`
ĉ
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
𝜕𝑇
e
− 𝑘,_
+ 𝑘,,
+ +𝑘,`
d̂ − 𝑘`_
+ 𝑘`,
+ +𝑘``
𝒌
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝜕𝑥
𝜕𝑦
𝜕𝑧
𝒒^^
%%
c(
c)%%
• conduction heat flux in a specific coordinate direction also due to temperature
gradients in other directions.
28
Thermal Conductivity (Cont.’d)
Range of thermal conductivity of various substances at normal pressure.
Some typical values:
𝑘AD = 350 − 385 𝑊/𝑚𝐾
𝑘l] = 8 − 10 𝑊/𝑚𝐾
𝑘]U[++VCCU (EF+DU[/ECF) = 0.038 𝑊/𝑚𝐾
𝑘N] = 360 − 410 𝑊/𝑚𝐾
𝑘VEFZCV ]U[++ = 0.78 𝑊/𝑚𝐾
𝑘[EW = 0.025 𝑊/𝑚𝐾
29
Thermal Conductivity (Cont.’d)
Variation of thermal
conductivity of
various substances
with temperature
30
Thermal Diffusivity
𝑘
𝛼=
𝜌𝑐
𝛼 = 𝑚 ! /𝑠
• Measure of the ability to conduct heat relative to the ability to
store heat
• As 𝛼 ↑, medium will respond more quickly to changes
(disturbances) à thermal equilibrium will be reached faster
31
Newton’s Law of Cooling
• Will use this as a boundary condition in heat conduction problems
Conduction in fluid on the wall
all
and
Newton’s Law of Cooling
(empirical)
thermal boundary layer
velocity boundary layer
Note that from energy balance (1st Law) on the wall:
Definition of heat transfer coefficient, h:
all
s
s
all
what is conducted to the wall
from the solid medium is
convected out to the fluid (B.C.
for the conduction medium)
all
32
Newton’s Law of Cooling (Cont.’d)
• h: a complex function of flow conditions, geometry and
thermophysical properties (determination largely
dependent on experimentation)
*
*
* contains latent heat storage (or release) due to phase change (in addition to
sensible heat storage (or release))
33
Stefan-Boltzmann Law of Radiation
• Will use this as a boundary condition in heat conduction problems
Radiation impinging on a surface.
• reflectivity ρ : fraction reflected back
• absorptivity α : fraction absorbed
• transmissivity τ : fraction transmitted thru
If opaque substance, τ = 0 à
Blackbody : ideal body which absorbs all impinging radiation energy without
reflection and transmission à α = 1; also emits maximum possible thermal
radiation
Blackbody emission:
Stefan-Boltzmann Law of Radiation
σ :Stefan–Boltzmann constant with the value, 5.67 × 10–8 W/(m2·K4)
34
Stefan-Boltzmann Law of Radiation (Cont.’d)
Real bodies (surfaces): emit radiation at a lower rate than a blackbody
Real surface emission:
𝜖
𝜖: emissivity of the surface (< 1)
Surface of interest with an emissivity 𝜖 and temperature 𝑇+ , exchanging heat only
by radiation with an environment modeled as a large enclosure, completely
enclosing the surface of interest at an effective blackbody temperature 𝑇X :
Net radiation flux from the (real) surface: 𝑞W^^ = 𝜖𝜎(𝑇+m − 𝑇Xm)
(temperature is absolute (in K))
• Occasionally, convenient to define a radiation heat transfer coefficient, hr
𝑞W^^ = ℎW (𝑇+ − 𝑇X )
where ℎW = 𝜖𝜎(𝑇+ + 𝑇X )(𝑇+R + 𝑇XR)
35
Stefan-Boltzmann Law of Radiation (Cont.’d)
environment, 𝑇$
𝑇& , 𝜖
G
𝒏
fluid, 𝑇%
𝑘, 𝑇
Energy balance (1st law) on the surface of a conduction medium exposed to
convection and radiation:
^^
𝑞YCFZ
=
^^
𝑞YCF\
^^
+ 𝑞W[Z
𝜕𝑇
→ −𝑘
𝜕𝑛
+DWT[YX
= ℎ(𝑇+ − 𝑇q) + 𝜖𝜎(𝑇+m − 𝑇Xm)
(B.C. for the conduction medium)
36