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Heat Transfer
Heat transfer and its applications.


Practically all the operations carried out by chemical involve the
production or absorption of energy in the form of heat .
The laws governing the transfer of heat and the type of apparatus that
have for their main object the control of heat flow are therefore of
great importance.
Examples are ubiquitous:
•heat flows in the body
•home heating/cooling systems
•refrigerators, ovens, other appliances
•automobiles, power plants, the sun,
etc.
Heat

A form of energy associated with the motion of atoms or molecules.

Transferred from higher temperature objects to objects at a lower
temperature.
The calorie had been defined as the amount of heat it takes to raise the
temperature of 1 gram of water by 1 degree C.
James Prescott Joule use his device below to find out how much work you
would have to do to create a calorie of heat.
Work done by
falling weights =
mgh
The Mechanical
Equivalent of Heat was
found to be 4.2 Joules of
mechanical work per
calorie of heat produced
4.2 J/cal
A 10 kg cinder block is dropped 50 meters. How many calories of heat will
it develop if dropped into 1000 kg water?
SAT2: How much will
the water’s temperature
go up?
Pool of water
Pool of water
Pool of water
Heat Transfer
There are 3 ways that heat can move from one place to another:
radiation
conduction
convection
Modes of Heat Transfer



Conduction - diffusion of heat due to temperature gradient
Convection - when heat is carried away by moving fluid
Radiation - emission of energy by electromagnetic waves
qconvection
qradiation
qconduction
Typical Design Problems

To determine:




overall heat transfer coefficient - e.g., for a car radiator
highest (or lowest) temperature in a system - e.g., in a gas turbine
temperature distribution (related to thermal stress) - e.g., in the walls of a
spacecraft
temperature response in time dependent heating/cooling problems - e.g.,
how long does it take to cool down a case of soda?
Conduction Heat Transfer


Conduction is the transfer of heat by molecular interaction
In a gas, molecular velocity depends on temperature


hot, energetic molecules collide with neighbors, increasing their speed
In solids, the molecules and the lattice structure vibrate
II. Unsteady-state conduction
T is a function of both location and
time: T=f(x,t)
III Steady state conduction
T is only function of location,
constant temperature distribution
H = the Rate of Heat Flow through a conductor
Unit:
H = Q/T = k A T
Thermal Conductivity
Temperature
difference
d
thickness
Cross-sectional area
Joules/sec or
Watts
Fourier’s Law

“heat flux is proportional to temperature gradient”
 T T 
Q
 q  kT  k 


A
 x y 


units for q
are W/m2
where k = thermal conductivity
in general, k = k(x,y,z,T,…)
temperature profile
heat conduction in a slab
dT
dx
1
hot wall
cold wall
x
Thermal Properties
Thermal Conductivity (k)
It is the term used to indicate the amount of heat that will pass through a unit of
area of a material at a temperature difference of one degree.
 The lower the “k” value, the better the insulation qualities of the material.
Units; US: (Btu.in) / (h.ft2.oF)

Metric: W / (m.oC)
Conductance (c)
It indicates the amount of heat that passes through a given thickness of material;
 Conductance= thermal conductivity / thickness
Units; US: Btu / (h.ft2.oF)
Metric: W/ (m2.oC)

Example:
Determination of Thermal Conductivity Coefficient for
Different Wall Systems (TS EN ISO 8990)
1
2
3
4
5
6
7
8
9
Cold Chamber
Freeze Fan
Thermo- Couples (3 unit) [cold chamber]
Thermo- couples (9 unit) [Surface]
Wall specimen (1.2 x 1.2 m)
Thermo- couples (9 unit) [Surface]
Hot Chamber
Thermo- Couples (3 unit) [hot chamber]
Heather Fan
Thermal Resistance (RSI for metric unit, R for US units)
 It is that property of a material that resist the flow of heat
through the material. It is the reciprocal of conductance;
R= 1/c
Thermal Transmittance (U)
 It is the amount of heat that passes through all the materials in a
system. It is the reciprocal of the total resistance;
U= 1/Rt

Table 1 lists a few of the common materials and their thermal
properties;
Table 6.1 Thermal properties of
materials
Brick, clay, 4 in (100 mm)
Built-up roofing
Concrete block, 8 in (200 mm):
Cinder
Lightweight aggregate
Glass, clear, ¼ in (6 mm)
Gypsum sheating, ½ in (12.5 mm)
Insulation, per 1 in (25 mm):
Fiberboard
Glass Fiber
Expanded Polystyrene
Rigid urethane
Vermiculite
Wood shavings
Moving air
Particle board, ½ in (12.5 mm)
Plywood, softwood, ¾ in (19 mm)
Stucco, ¾ in (19 mm)
Thermal Resistance
RSI
R
0.07
0.42
0.08
0.44
a
Thermal Conductivity
K (SI)
K (US customary)
1.43
9.52
0.30
0.35
0.16
0.08
1.72
2.00
0.91
0.43
0.67
0.57
0.04
0.16
4.65
4.00
0.27
1.16
0.49
0.52
0.75
1.05
0.36
0.42
0.03
0.11
0.17
0.02
2.80
2.95
4.23
6.00
2.08
2.44
0.17
0.62
0.97
0.11
0.051
0.048
0.033
0.024
0.069
0.060
0.36
0.34
0.24
0.17
0.48
0.41
0.114
0.112
0.95
0.81
0.77
6.82
Variable Thermal Conductivity, k(T)



The thermal conductivity of a
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
for other temperaturedependent properties such as
the density and specific heat.
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 the temperature range between T1 and T2
can be determined from
T
kave


2
T1
k (T )dT
(2-75)
T2  T1
The variation in thermal conductivity of a material
with can often be approximated as a linear function
and expressed as
k (T )  k0 (1   T )
(2-79)
 the temperature coefficient of thermal conductivity.
Variable Thermal Conductivity


For a plane wall the
temperature varies linearly
during steady onedimensional heat conduction
when the thermal conductivity
is constant.
This is no longer the case
when the thermal conductivity
changes with temperature
(even linearly).
Let’s try a sample problem using:
H = Q/T = k A T
d
A steel slab 5 cm thick is used as a firewall, measuring 3 m x 4 m. If a fire burns at
800 C on one side of a wall, how fast will heat flow through the metal door. (The
conductivity of steel is 46 Watts/m•K)
Fourier’s Law
and the
Heat Equation
Fourier’s Law
Fourier’s Law
• A rate equation that allows determination of the conduction heat flux
from knowledge of the temperature distribution in a medium
• Its most general (vector) form for multidimensional conduction is:


q   k  T
Implications:
– Heat transfer is in the direction of decreasing temperature
(basis for minus sign).
– Fourier’s Law serves to define the thermal conductivity of the
 
medium  k   q/  T 




– Direction of heat transfer is perpendicular to lines of constant
temperature (isotherms).
– Heat flux vector may be resolved into orthogonal components.
Heat Flux Components
• Cartesian Coordinates: T  x, y, z 

T  T  T 
q  k
i k
jk
k
x
y
z
qx
qz
qy
(2.3)
• Cylindrical Coordinates: T  r ,  , z 

T 
T  T 
q   k
i k
jk
k
r
r 
z
qr
q
(2.18)
qz
• Spherical Coordinates: T  r ,  , 


T 
T 
T
q  k
i k
jk
k
r
r 
r sin  
q
q
qr
(2.21)
Heat Flux Components (cont.)
• In angular coordinates  or  ,  , the temperature gradient is still
based on temperature change over a length scale and hence has
units of C/m and not C/deg.
• Heat rate for one-dimensional, radial conduction in a cylinder or sphere:
– Cylinder
qr  Ar qr  2 rLqr
or,
qr  Ar qr  2 rqr
– Sphere
qr  Ar qr  4 r 2 qr
Heat Equation
The Heat Equation
• A differential equation whose solution provides the temperature distribution in
a stationary medium.
• Based on applying conservation of energy to a differential control volume
through which energy transfer is exclusively by conduction.
• Cartesian Coordinates:
  T
k
x  x
   T
  k
 y  y
   T
  k
 z  z
Net transfer of thermal energy into the
control volume (inflow-outflow)
T
 •
  q  c p
t

Thermal energy
generation
(2.13)
Change in thermal
energy storage
Generalized Heat Diffusion Equation

If we perform a heat balance on a small volume of material…
heat conduction
in

T
q
… we get:
heat conduction
out
heat generation
T
2
c
 k T  q
t
rate of change
of temperature

heat cond. heat
in/out
generation
k
 thermal diffusivity
c
Heat Equation (Radial Systems)
• Cylindrical Coordinates:
1   T  1   T    T  •
T
kr

k

k

q


c
p






r r  r  r 2     z  z 
t
(2.20)
• Spherical Coordinates:
1   2 T 
1
  T 
1
 
T  •
T
kr

k

k
sin


q


c
p






r  r 2 sin 2      r 2 sin   
 
t
r 2 r 
(2.33)
Heat Equation (Special Case)
• One-Dimensional Conduction in a Planar Medium with Constant Properties
and No Generation
 2T
x 2


1 T
 t
k
 thermal diffusivity of the medium
c p
Boundary Conditions
Boundary and Initial Conditions
• For transient conduction, heat equation is first order in time, requiring
specification of an initial temperature distribution: T  x, t t 0  T  x,0
• Since heat equation is second order in space, two boundary conditions
must be specified. Some common cases:
Constant Surface Temperature:
T  0, t   Ts
Constant Heat Flux:
Applied Flux
Insulated Surface
k
T
|x  0  qs
x
k
T
|x  0  h T  T  0, t  
x
Convection
T
|x  0  0
x
Boundary Conditions

Heat transfer boundary conditions generally come in three types:
q = 20 W/m2
specified heat flux
Neumann condition
T = 300K
specified temperature
Dirichlet condition
Tbody
q = h(Tamb-Tbody)
external heat transfer
coefficient
Robin condition
Properties
Thermophysical Properties
Thermal Conductivity: A measure of a material’s ability to transfer thermal
energy by conduction.
Thermal Diffusivity: A measure of a material’s ability to respond to changes
in its thermal environment.
Property Tables:
Solids: Tables A.1 – A.3
Gases: Table A.4
Liquids: Tables A.5 – A.7
Conduction Analysis
Methodology of a Conduction Analysis
• Solve appropriate form of heat equation to obtain the temperature
distribution.
• Knowing the temperature distribution, apply Fourier’s Law to obtain the
heat flux at any time, location and direction of interest.
• Applications:
Chapter 3: One-Dimensional, Steady-State Conduction
Chapter 4: Two-Dimensional, Steady-State Conduction
Chapter 5: Transient Conduction
Chapter 2: Heat Conduction
Equation
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
 Understand multidimensionality and time dependence of heat transfer,
and the conditions under which a heat transfer problem can be
approximated as being one-dimensional,
 Obtain the differential equation of heat conduction in various
coordinate systems, and simplify it for steady one-dimensional case,
 Identify the thermal conditions on surfaces, and express them
mathematically as boundary and initial conditions,
 Solve one-dimensional heat conduction problems and obtain the
temperature distributions within a medium and the heat flux,
 Analyze one-dimensional heat conduction in solids that involve heat
generation, and
 Evaluate heat conduction in solids with temperature-dependent
thermal conductivity.
Introduction




Although heat transfer and temperature are closely related, they are of a
different nature.
Temperature has only magnitude
it is a scalar quantity.
Heat transfer has direction as well as magnitude
it is a vector quantity.
We work with a coordinate system and indicate direction with plus or minus
signs.
Introduction ─ Continue



The driving force for any form of heat transfer is the
temperature difference.
The larger the temperature difference, the larger the
rate of heat transfer.
Three prime coordinate systems:



rectangular (T(x, y, z, t)) ,
cylindrical (T(r, , z, t)),
spherical (T(r, , , t)).
Introduction ─ Continue



Classification of conduction heat transfer problems:
steady versus transient heat transfer,
multidimensional heat transfer,
heat generation.
Steady versus Transient Heat Transfer

Steady implies no change with time at any point
within the medium

Transient implies variation with time or time
dependence
Multidimensional Heat Transfer

Heat transfer problems are also classified as being:




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-dimensional
depending on the relative magnitudes of heat transfer
rates in different directions and the level of accuracy
desired.

The rate of heat conduction through a medium in
a specified direction (say, in the x-direction) is
expressed by Fourier’s law of heat conduction
for one-dimensional heat conduction as:
Qcond

dT
 kA
dx
(W) (2-1)
Heat is conducted in the direction
of decreasing temperature, and thus
the temperature gradient is negative
when heat is conducted in the positive xdirection.
General Relation for Fourier’s Law of
Heat Conduction


The heat flux vector at a point P on the surface of
the figure must be perpendicular to the surface,
and it must point in the direction of decreasing
temperature
If n is the normal of the
isothermal surface at point P,
the rate of heat conduction at
that point can be expressed by
Fourier’s law as
dT
Qn  kA
(W) (2-2)
dn
General Relation for Fourier’s Law of
Heat Conduction-Continue

In rectangular coordinates, the heat conduction
vector can be expressed in terms of its components as
Qn  Qx i  Qy j  Qz k

(2-3)
which can be determined from Fourier’s law as

T
Qx  kAx x

T

Qy  kAy
y


T
Qz  kAz
z

(2-4)
Heat Generation

Examples:







electrical energy being converted to heat at a rate of I2R,
fuel elements of nuclear reactors,
exothermic chemical reactions.
Heat generation is a volumetric phenomenon.
The rate of heat generation units : W/m3 or Btu/h · ft3.
The rate of heat generation in a medium may vary
with time as well as position within the medium.
The total rate of heat generation in a medium of
volume V can be determined from
Egen   egen dV
V
(W)
(2-5)
One-Dimensional Heat Conduction
Equation - Plane Wall
Rate of heat Rate of heat
conduction - conduction +
at x
at x+x
Qx Qx x  Egen,element
(2-6)
Rate of heat
generation
inside the
element
=
Eelement

t
Rate of change of
the energy
content of the
element
Qx  Qx x  Egen ,element

Eelement

t
(2-6)
The change in the energy content and the rate of heat
generation can be expressed as

 Eelement  Et t  Et  mc Tt t  Tt    cAx Tt t  Tt  (2-7)

(2-8)
 Egen,element  egenVelement  egen Ax

Substituting into Eq. 2–6, we get
Qx  Qx x egen Ax   cAx
Tt t  Tt
t
(2-9)
• Dividing by Ax, taking the limit as x 0 and t 0,
and from Fourier’s law:
1   T
 kA
A x  x
T


e


c
 gen
t

(2-11)
The area A is constant for a plane wall  the one dimensional
transient heat conduction equation in a plane wall is
Variable conductivity:
  T
k
x  x
T

  egen   c
t

Constant conductivity:
 2T egen 1 T


2
x
k
 t
; 
(2-13)
k
c
(2-14)
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
1) Steady-state:
d 2T egen

0
2
dx
k
(2-15)
2) Transient, no heat generation:
 2T 1 T

2
x
 t
(2-16)
d 2T
0
2
dx
(2-17)
3) Steady-state, no heat generation:
One-Dimensional Heat Conduction
Equation - Long Cylinder
Rate of heat Rate of heat
conduction - conduction +
at r
at r+r
Qr Qr r  Egen,element
Rate of heat
generation
inside the
element
=
Eelement

t
(2-18)
Rate of change of
the energy
content of the
element
Qr  Qr r  Egen ,element
Eelement

t
(2-18)
• The change in the energy content and the rate of heat
generation can be expressed as

 Eelement  Et t  Et  mc Tt t  Tt    cAr Tt t  Tt  (2-19)

(2-20)
 Egen,element  egenVelement  egen Ar
• Substituting into Eq. 2–18, we get
Qr  Qr r
 egen Ar   cAr
Tt t  Tt
t
(2-21)
• Dividing by Ar, taking the limit as r 0 and t 0,
and from Fourier’s law:
1   T
 kA
A r  r
T


e


c
 gen
t

(2-23)
Noting that the area varies with the independent variable r
according to A=2rL, the one dimensional transient heat
conduction equation in a plane wall becomes
Variable conductivity:
Constant conductivity:
1   T
 rk
r r  r
T


e


c
 gen
t

1   T  egen 1 T

r

r r  r  k
 t
(2-25)
(2-26)
The one-dimensional conduction equation may be reduces
to the following forms under special conditions
1) Steady-state:
2) Transient, no heat generation:
1 d  dT  egen
 0 (2-27)
r

r dr  dr  k
1   T
r
r r  r
3) Steady-state, no heat generation:
 1 T

  t
d  dT
r
dr  dr

0

(2-28)
(2-29)
One-Dimensional Heat Conduction
Equation - Sphere
Variable conductivity:
1   2 T
r k
2
r r 
r
T

  egen   c
t

(2-30)
Constant conductivity:
1   2 T  egen 1 T

r

2
r r  r  k
 t
(2-31)
General Heat Conduction Equation
Rate of heat Rate of heat Rate of heat Rate of change
conduction - conduction + generation = of the energy
at x, y, and z
inside the
content of the
at x+x,
element
element
y+y, and
z+z
Qx  Qy  Qz Qxx  Qy y  Qz z
 Egen ,element 
Eelement
(2-36)
t
Repeating the mathematical approach used for the onedimensional heat conduction the three-dimensional heat
conduction equation is determined to be
Two-dimensional
Constant conductivity:
 2T  2T  2T egen 1 T
 2  2 

2
x
y
z
k
 t
(2-39)
Threedimensional
1) Steady-state:
 2T  2T  2T egen
 2 2 
 0 (2-40)
2
x
y
z
k
2) Transient, no heat generation:
 2T  2T  2T 1 T
 2  2 
(2-41)
2
x
y
z
 t
3) Steady-state, no heat
 2T  2T  2T
generation: 2  2  2  0
x
y
z
(2-42)
Cylindrical Coordinates
1   T
 rk
r r  r
 1 T  T    T
k
 k
 2
 r     z  z
T

  egen   c
t

(2-43)
Spherical Coordinates
1   2 T
 kr
2
r r 
r
1
  T 
1
 
T 
T

k
 2
 2 2
 k sin 
  egen   c
 
t
 r sin      r sin   
(2-44)
Boundary and Initial Conditions






Specified Temperature Boundary Condition
Specified Heat Flux Boundary Condition
Convection Boundary Condition
Radiation Boundary Condition
Interface Boundary Conditions
Generalized Boundary Conditions
Specified Temperature Boundary
Condition
For one-dimensional heat transfer
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) = T2
(2-46)
The specified temperatures can be constant, which is the
case for steady heat conduction, or may vary with time.
Specified Heat Flux Boundary
Condition
The heat flux in the positive xdirection anywhere in the medium,
including the boundaries, can be
expressed by Fourier’s law of heat
conduction as
dT
q  k

dx
Heat flux in the
positive xdirection
(2-47)
The 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.
Two Special Cases
Insulated boundary
T (0, t )
k
0
x
or
T (0, t )
0
x
(2-49)
Thermal symmetry


T L , t
2
0
x
(2-50)
Convection Boundary Condition
Heat conduction
at the surface in
a
selected
direction
and
=
Heat convection
at the surface in
the same
direction
T (0, t )
k
 h1 T1  T (0, t ) 
x
T ( L, t )
k
 h2 T ( L, t )  T 2 
x
(2-51a)
(2-51b)
Radiation Boundary Condition
Heat conduction
at the surface in
a
selected
direction
and
=
Radiation
exchange at the
surface in
the same
direction
T (0, t )
4
4

k
 1 Tsurr

T
(0,
t
)
,1
x
(2-52a)
T ( L, t )
4

k
  2 T ( L, t ) 4  Tsurr
,2 
x
(2-52b)
Interface Boundary Conditions
At the interface the requirements are:
(1) two bodies in contact must have the same
temperature at the area of contact,
(2) an interface (which is a
surface) cannot store any
energy, and thus the heat flux
on the two sides of an
interface must be the same.
TA(x0, t) = TB(x0, t)
and
k A
(2-53)
TA ( x0 , t )
T ( x , t )
 k B B 0 (2-54)
x
x
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
Heat transfer
to the surface
in all modes
=
Heat transfer
from the surface
In all modes
Heat Generation in Solids
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.
Heat Generation in Solids -The Surface
Temperature
Rate of
heat transfer
from the solid
=
Rate of
energy
generation
within the solid
(2-63)
For uniform heat generation within the medium
Q  egenV (W)
(2-64)
The heat transfer rate by convection can also be
expressed from Newton’s law of cooling as
-
Q  hAs Ts  T 
Ts  T 
(W)
egenV
hAs
(2-65)
(2-66)
Heat Generation in Solids -The Surface
Temperature
For a large plane wall of thickness 2L (As=2Awall and
V=2LAwall)
egen L
(2-67)
Ts , plane wall  T 
h
For a long solid cylinder of radius r0 (As=2r0L and
V=r02L)
egen r0
(2-68)
Ts ,cylinder  T 
2h
For a solid sphere of radius r0 (As=4r02 and V=4/3r03)
Ts , sphere  T 
egen r0
3h
(2-69)
Heat Generation in Solids -The maximum
Temperature in a Cylinder (the Centerline)
The heat generated within an inner
cylinder must be equal to the heat
conducted through its outer surface.
kAr
dT
 egenVr
dr
(2-70)
Substituting these expressions into the above equation
and separating the variables, we get


egen
dT
2
k  2 rL 
 egen  r L  dT  
rdr
dr
2k
Integrating from r =0 where T(0) =T0 to r=ro
Tmax,cylinder  T0  Ts 
egen r02
4k
(2-71)
Variable Thermal Conductivity, k(T)



The thermal conductivity of a
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
for other temperaturedependent properties such as
the density and specific heat.
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 the temperature range between T1 and T2
can be determined from
T
kave


2
T1
k (T )dT
(2-75)
T2  T1
The variation in thermal conductivity of a material
with can often be approximated as a linear function
and expressed as
k (T )  k0 (1   T )
(2-79)
 the temperature coefficient of thermal conductivity.
Variable Thermal Conductivity


For a plane wall the
temperature varies linearly
during steady onedimensional heat conduction
when the thermal conductivity
is constant.
This is no longer the case
when the thermal conductivity
changes with temperature
(even linearly).