Principles of Chemical Engineering
Chapter 4. Heat Transfer
Yanwei Wang (王衍伟)*
Nan Fu (傅楠)*
Department of Polymer Science and
Engineering
College of Chemistry, Chemical
Engineering and Materials science
苏州大学材化部高分子科学与工程系
*Email: [email protected].
Office: 907-1427
Chemical Engineering Innovation
Laboratory
College of Chemistry, Chemical
Engineering and Materials science
苏州大学材化部化工创新实验室
*Email: [email protected].
Office: 701-1517
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What is heat?
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Heat is energy.
Heat is always associated with
a process of some kind.
Heat naturally flows from
hotter to colder systems.
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Common Temperature Scales
Hot
Cold
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Conversion between
temperature units
[°C] = ( [°F] – 32 )(5/9)
[°K] = ( [°C] + 273.15 )
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Zeroth Law of Thermodynamics
If bodies A and B are each in thermal equilibrium with a
third body C, then they are in thermal equilibrium with
each other.
Translation: Every body has a property called temperature.
When two bodies (one might be a thermometer) are
found to be in thermal equilibrium, then their
temperatures are the same. This may be used to
determine the temperature of a third body (through
calibration).
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DEFINITION OF HEAT
Heat is energy that flows from a
higher temperature object to a lower
temperature object because of a
difference in temperatures.
SI Unit of Heat: joule (J)
OTHER UNITS
1 kcal = 4186 joules
1 cal = 4.186 joules
1 BTU = 1055 J
The heat that flows from
hot to cold originates in
the internal energy of
the hot substance.
It is NOT correct to say that
a substance
contains heat.
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Heat and Temperature Change: Specific Heat Capacity
The heat that must be supplied or removed to
change the temperature of a substance is
Q  mc p T
specific heat
capacity
Common Unit for Specific
Heat Capacity: J/(kg·K)
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Exercise:
When you drink cold water, your body must expend metabolic
energy to maintain normal body temperature of 37 oC by
warming up the water in your stomach. Could drinking ice
water substitute for exercise as a way to “burn calories?”
Suppose you expend 430 kilocalories
during a brisk one-hour walk. How
many liters of ice water would you
have to drink in order to use 430
kilocalories of metabolic energy?
Note: the stomach can hold about one
liter.
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The three modes of heat
transfer
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Heat conduction
• Heat conduction is a mode of
transfer of energy due to a
temperature gradient.
• Heat flows from the region of higher
temperature to that at a lower
temperature unaccompanied by any
observable macroscopic motion.
• Conduction is a microscopic–level
mechanism, resulting from the
exchange of translational, rotational,
and vibrational energy among the
molecules comprising the medium.
• In metals, heat conduction results
from the motion of free electrons.
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Convection
Convection is the process in which heat is carried from one
place to another by the bulk movement of a fluid.
Natural convection: Movement of a fluid is due to buoyant forces created by density
differences, and the density differences are caused by temperature gradients in the fluid.
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Thermal Radiation
RADIATION
A process in which
energetic particles or
energetic waves travel
through a vacuum, or
through mattercontaining media that
are not required for
their propagation.
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Heat Transfer by
Conduction
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Heat conduction
• In solids that are poor conductors
of electricity and in most liquids,
heat conduction results from
momentum transfer between
adjacent vibrating molecules or
atoms.
• In gases, heat conduction occurs
by the random motion of
molecules, so heat is diffused
from hotter regions to colder ones.
• Check Appendices 9-12 for
thermal conductivity data.
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Fourier’s law for heat conduction
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Thermal conductivity
The magnitude of thermal conductivity spans five orders of magnitude.
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Steady state heat conduction
•
steady flow energy equation

q0
all surfaces
q : rate of heat flow in direction normal to surface.
•
For the 1D case shown in the figure, the heat
transfer rate in at the left (at x) is
q( x  dx)
q ( x)
 dT 
q( x)    kA

 dx  x
•
The heat transfer rate on the right is
q( x  dx)  q( x) 
•
Hence,
q( x  dx)  q( x)  0
dq
dx  
dx x

(Taylor expansion)
dq d  dT
  kA
dx dx  dx
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
0

A and k may be
functions of x.
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Thermal Resistance Circuits
R = B/K is the thermal resistance ((m2·K)/W).
1/R – Thermal conductance (heat transfer coefficient h (W/(m2·K))).
Rth = B/(kA) is the absolute thermal resistance (K/W).
Thermal resistors in series
TL  TR TL  TR
q
=
Rtotal
R1  R2
Thermal resistors in Parallel
q  q1  q2  q3
1
Rtotal
 R11  R21  R31
q
q
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Example in the textbook
RA  BA  k A A 
RB  BB
RC  BC
 kB A
 kC A 
TA TB TC
q


RA
RB
RC
T  TA  TB  TC  q  RA  RB  RC 
q
TA =
BA
kA A
Requirement:
Understand and be able to draw the
temperature profile.
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Supplementary - 1
Fourier’s law in cylindrical and spherical coordinates
Fourier's law: j   k T
T
T
T
xˆ 
yˆ 
zˆ
x
y
z
where xˆ, yˆ , zˆ are the unit vectors in their respective directions.
In Cartesian coordinates ( x, y, z ) : T 
T
1 T ˆ T
In Cylindrical coordinates ( r ,  , z ) : T 
rˆ 

zˆ
r
r 
z
where rˆ, ˆ, zˆ are the unit vectors in their respective directions.
In Spherical coordinates ( r ,  ,  ) : T 
T
1 T ˆ
1 T ˆ
rˆ 


r
r 
r sin  
where rˆ, ˆ, ˆ are the unit vectors in their respective directions.
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Sample derivation (for the cylindrical case)
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T in cylindrical coordinates (r ,  , z )
 xˆ   cos   rˆ    sin   ˆ

&
 yˆ   sin   rˆ   cos  ˆ
 zˆ  zˆ

rˆ   cos   xˆ   sin   yˆ
 ˆ
    sin   xˆ   cos   yˆ
 zˆ  zˆ

 x  r cos 

 y  r sin 
z  z

r  x 2  y 2

  arctan  y x 
z  z

T T
 x  r

 T T



y
r



 
 z z
r T

x 
r T

y 
&

T  sin  T
 cos 

x
r
r 

T cos  T
 sin 

y
r
r 
&
 T T
 r  x

 T T




x



 
 z z
x T y
T
T

 cos 
 sin 
r y r
x
y
x T y
T
T

   r sin  
  r cos  
 y 
x
y
T
T
T
T  sin  T  

ˆ
xˆ 
yˆ 
zˆ   cos 

  cos   rˆ    sin   
x
y
z
r
r  

T cos  T  

ˆ T zˆ  T rˆ  1 T ˆ  T zˆ
  sin 

  sin   rˆ   cos    
r
r  
z
r
r 
z

T 
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Example-1: Heat conduction through a plane slab
Steady state heat conduction
d  dT 
 kA
0
dx  dx 
For constant k and A, we have
dT
 C1  T  C1 x  C2
dx
where C1 and C2 are constants of integration.
Boundary conditions: T  T1 at x  x1
T  T2 at x  x2

T  T1
x  x1

T2  T1 x2  x1
Temperature
distribution through
the thin slab.
Rate of heat flow
T T
T1  T2
dT
q  kA
 kA 1 2 
dx
x2  x1  x2  x1   kA 
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Note:
• the linear temperature profile
• the direction of heat flow
(from regions of high
temperature to regions of low
temperature.)
• the analogy with Ohm’s law
and Poiseuille’s law
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Steady Quasi-One-Dimensional Heat Flow
in Non-Planar Geometry
Heat is lost from a hot-water pipe
to the air outside in the radial
direction.
Heat transfer from a long pipe is
one dimensional
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Steady heat flow through a cylindrical shell
q
dT
 k
A
dr
q  2rL(k
dT
T T
k AL (T1  T2 )
)  2rLk 1 2 
dr
ln( r2 / r1 )
(r2  r1 )
r2  r1
where AL  2 r L L, and r L 
ln( r2 / r1 )
Heat transfer q  k T
B
through a thin A
slab
kA(T1  T2 )
q
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( x2  x1 )
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Logarithmic mean radius
The logarithmic mean is a function of two non-negative numbers which is
equal to their difference divided by the logarithm of their quotient. The
logarithmic mean of two numbers is smaller than the arithmetic mean but
larger than the geometric mean (unless the numbers are the same, in which
case all three means are equal to the numbers).
rg  rL  ra  rrms
When r0/ri is close to
1, rL approximately
equals ra.
logarithmic mean: rL   ro  ri  ln  ro ri 
arithmetic mean: ra   ro  ri  2
geometric mean: rg  ro ri
root mean square: rrms 
2
2
r

r
o i  2
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Steady heat flow through a cylindrical shell
d  dT 
 kA
  0 , A  2 rL
dr  dr 
dT
r
 C1 where C1 is a constant of integration.
dr
d  r r1 
 dT  C1
 T  C1 ln  r r1   C2
r r1
Boundary conditions: T  T1 at r  r1 , T  T2 at r  r2

T  T1 ln  r r1 

T2  T1 ln  r2 r1 
Temperature distribution
through the cross-section of
the cylindrical shell.
Heat transfer
through a thin T  T1  x  x1
T2  T1 x2  x1
slab
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Exercise:
A tube of 60-mm outer diameter (OD) is
insulated with a 50-mm layer of silica foam, for
which the conductivity is 0.055 W/(m·°C),
followed with a 40-mm layer of cork with a
conductivity of 0.05 W/(m·°C). If the
temperature of the outer surface of the pipe is
150°C and the temperature of the outer
surface of the cork is 30oC, calculate the heat
loss in watts per meter of pipe.
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Supplementary-2: The heat
equation
Equation
of change for
temperature, in terms of the heat
flux vector q and the viscous
(for unsteady heat conduction)
momentum flux tensor τ
^
 C p is the volumetric specific
heat capacity, describing the
ability to store internal energy
without phase transition.
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Derivation of the heat equation (1/2)
• The heat equation is derived based on the conservation of energy and
Fourier’s law.
• For simplicity, we assume
– temperature is uniform over cross-sections (one-dimensional)
– Heat transfer is only by conduction
– No heat escapes from sides (perfect insulation)
x
  Ax  c p T   q( x, t )  q( x  x, t )  t
T ( x, t ): temperature
q ( x, t ): rate of heat flow
c p : specific heat capacity
q ( x, t )
T ( x, t )  T ( x  x, t )
 k
A
x
: density
q ( x  x, t )
T ( x  x, t )  T ( x, t )
 k
A
x
k: thermal conductivity
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Derivation of the heat equation (1/2)
  Ax  c p T   q( x, t )  q( x  x, t ) t
  Ax  c p T   kA

T ( x, t )  T ( x  x, t )
T ( x  x, t )  T ( x, t ) 
 kA
t

x
x

T ( x  x, t )  T ( x, t ) T ( x, t )  T ( x  x, t )

T
k
x
x

t  c p
x
T
k  2T

t  c p x 2

(for constant k )
T
k
2
  T where  =
t
cp
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Heat Transfer by
Conduction
Summary
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• The relationship between temperature and heat
• Three concepts:
– Specific heat capacity
– Thermal conductivity
– Thermal resistance (circuits)
• Definition - heat conduction
• Calculation - Fourier’s law
• Steady state heat conduction in thin slab and cylindrical
shell
• Logarithmic mean radius
• Supplementary
– The heat equation (for unsteady heat conduction)
– Transformation between Cartesian coordinates and Cylindrical
coordinates
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