chapter 3: steady heat conduction -...
TRANSCRIPT
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Chapter 3: Steady Heat Conduction
3-1 Steady Heat Conduction in Plane Walls
3-2 Thermal Resistance
3-3 Steady Heat Conduction in Cylinders
3-4 Steady Heat Conduction in Spherical Shell
3-5 Steady Heat Conduction with Energy Generation
3-6 Heat Transfer from Finned Surfaces
3-7 Bioheat Equation
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-1 Steady Heat Conduction in Plane Walls (1)
3-1
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-1 Steady Heat Conduction in Plane Walls (2)
0
0wallin out
dEQ Q
dt
, constantcond wallQ
Assuming heat transfer is the only energy interaction and
there is no heat generation, the energy balance can be expressed as
The rate of heat transfer through the wall
must be constant ( ).
1 2, (W)cond wall
T TQ kA
L
3-2
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (1)
1 2, (W)cond wall
wall
T TQ
R
( C/W)wall
LR
kA
where Rwall is the conduction resistance (or thermal resistance)
Analogy to Electrical Current Flow: 1 2
eR
V VI
Heat Transfer Electrical current flow
Rate of heat transfer Electric current
Thermal resistance Electrical resistance
Temperature difference Voltage difference
3-3
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (2)
Thermal resistance can also be applied to
convection processes.
Newton’s law of cooling for convection heat
transfer rate ( ) can be rearranged as
Rconv is the convection resistance
conv s sQ hA T T
(W)sconv
conv
T TQ
R
1 ( C/W)conv
s
RhA
3-4
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (2)
Radiation Resistance: The rate of radiation heat
transfer between a surface and the surrounding
4 4 ( ) (W)s surrrad s s surr rad s s surr
rad
T TQ A T T h A T T
R
1 ( /W)rad
rad s
R Kh A
2 2 2 (W/m K)( )
radrad s surr s surr
s s surr
Qh T T T T
A T T
3-5
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Radiation and Convection Resistance
3-2 Thermal Resistance (3)
A surface exposed to the surrounding might involves
convection and radiation simultaneously.
The convection and radiation resistances are parallel to each
other.
When Tsurr≈T∞, the radiation
effect can properly be
accounted for by replacing h
in the convection resistance
relation by
hcombined = hconv+hrad (W/m2K)
3-6
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (4)
Thermal Resistance Network
1 ,1 1
1 22 2 ,2
Q h A T T
T TkA h A T T
L
Rate of
heat convection
into the wall
Rate of
heat conduction
through the wall
Rate of
heat convection
from the wall = =
,1 ,2 (W)
total
T TQ
R
,1 ,2
1 2
1 1 ( C/W)total conv wall conv
LR R R R
h A kA h A
3-7
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (5)
It is sometimes convenient to express heat transfer
through a medium in an analogous manner to
Newton’s law of cooling as
where U is the overall heat transfer coefficient.
Note that
(W)Q UA T
1 ( C/K)
total
UAR
3-8
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (6)
Multilayer Plane Walls
,1 ,1 ,2 ,2
1 2
1 1 2 2
1 1
total conv wall wall convR R R R R
L L
h A k A k A h A
3-9
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Thermal Contact Resistance
3-2 Thermal Resistance (7)
-In reality surfaces have some roughness.
When two surfaces are pressed against each other,
the peaks form good material contact but the valleys
form voids filled with air.
-As a result, an interface contains
numerous air gaps of varying sizes
that act as insulation because of the
low thermal conductivity of air.
-Thus, an interface offers some
resistance to heat transfer, which
is termed the thermal contact
resistance, Rc. 3-10
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Generalized Thermal Resistance Network
3-2 Thermal Resistance (8)
1 2 1 21 2 1 2
1 2 1 2
1 1T T T TQ Q Q T T
R R R R
1
totalR
1 2
1 2 1 2
1 1 1 = total
total
R RR
R R R R R
3-11
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-2 Thermal Resistance (9)
Combined Series-Parallel
Arrangement
1
total
T TQ
R
1 212 3 3
1 2
total conv conv
R RR R R R R R
R R
31 21 2 3
1 1 2 2 3 3 3
1 ; ; ; conv
LL LR R R R
k A k A k A hA
3-12
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-3 Steady Heat Conduction in Cylinders (1)
Consider the long cylindrical layer
Assumptions:
the two surfaces of the cylindrical layer are maintained at
constant temperatures T1 and T2,
no heat generation,
constant thermal conductivity,
one-dimensional heat conduction.
Fourier’s law of heat conduction
, (W)cond cyl
dTQ kA
dr
3-13
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Separating the variables and integrating from r=r1, where
T(r1)=T1, to r=r2, where T(r2)=T2
Substituting A =2prL and performing the integrations give
Since the heat transfer rate is constant
3-3 Steady Heat Conduction in Cylinders (2)
, (W)cond cyl
dTQ kA
dr
2 2
1 1
,
r T
cond cyl
r r T T
Qdr kdT
A
1 2
,
2 1
2ln /
cond cyl
T TQ Lk
r r
1 2,cond cyl
cyl
T TQ
R
3-14
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Alternative method:
3-3 Steady Heat Conduction in Cylinders (3)
3-15
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-3 Steady Heat Conduction in Cylinders (3)
Thermal Resistance with Convection
,1 ,2
total
T TQ
R
Steady one-dimensional heat transfer through a
cylindrical or spherical layer that is exposed to
convection on both sides
where
,1 ,2
2 1
1 1 2 2
ln /1 1
2 2 2
total conv cyl convR R R R
r r
r L h Lk r L h
3-16
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-3 Steady Heat Conduction in Cylinders (4)
Multilayered Cylinders
,1 ,1 ,3 ,3 ,2
2 1 3 2 4 3
1 1 1 2 3 2 2
ln / ln / ln /1 1
2 2 2 2 2
total conv cyl cyl cyl convR R R R R R
r r r r r r
r L h Lk Lk Lk r L h
3-17
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-3 Steady Heat Conduction in Cylinders (5)
3-18
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Critical Radius of Insulation
1 1
2 1
2
ln / 1
2 2
ins conv
T T T TQ
r rR R
Lk h r L
2
0dQ
dr , (m)cr cylinder
kr
h
The value of r2 at which reaches a
maximum is determined by
3-3 Steady Heat Conduction in Cylinders (6)
Thus, insulating the pipe may actually
increase the rate of heat transfer instead of
decreasing it. 3-19
0d
2
2
2
dr
Q
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Critical Radius of Insulation
-Adding more insulation to a wall or to the attic always decreases heat transfer.
-Adding insulation to a cylindrical pipe or a spherical shell, however, is a different matter.
-Adding insulation increases the conduction resistance of the insulation layer but decreases the convection resistance of the surface because of the increase in the outer surface area for convection.
-The heat transfer from the pipe may increase or decrease, depending on which effect dominates.
3-3 Steady Heat Conduction in Cylinders (7)
3-20
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Spherical Shell
3-4 Steady Heat Conduction in Spherical Shell
3-21
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-5 Steady Heat Conduction with Energy Generation (1)
3-22
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-5 Steady Heat Conduction with Energy Generation (2)
L
TTL
k
qC
L
TTC
ssss
22,
2
1,2,2
2
1,2,
1
3-23
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-5 Steady Heat Conduction with Energy Generation (3)
0 gout EE
3-24
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-5 Steady Heat Conduction with Energy Generation (4)
Combined one-dimensional heat
conduction equation
n=0: plane wall (r–>x)
n=1: cylinder
n=2: sphere
3-25
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-5 Steady Heat Conduction with Energy Generation (5)
and
h
rqTTEE sgout
30 0
3-26
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-6 Heat Transfer from Finned Surfaces (1)
Newton’s law of cooling
Two ways to increase the rate of heat transfer:
increasing the heat transfer coefficient,
increase the surface area fins
Fins are the topic of this section.
conv s sQ hA T T
3-27
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Fin Equation
3-6 Heat Transfer from Finned Surfaces (2)
Under steady conditions, the energy balance on this
volume element can be expressed as
or
where
Substituting and dividing by x, we obtain
Rate of heat
conduction into
the element at x
Rate of heat
conduction from the
element at x+x
Rate of heat
convection from
the element
= +
, ,cond x cond x x convQ Q Q
convQ h p x T T
, ,0
cond x x cond xQ Qhp T T
x
3-28
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
0conddQhp T T
dx
cond c
dTQ kA
dx
0c
d dTkA hp T T
dx dx
22
20
dm
dx
; c
hpT T m
kA
1 2( ) mx mxx C e C e
3-6 Heat Transfer from Finned Surfaces (3)
3-29
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Boundary Conditions Several boundary conditions are typically employed:
At the fin base
Specified temperature boundary condition, expressed
as: q(0)= qb= Tb-T∞
At the fin tip 1. Specified temperature
2. Infinitely Long Fin
3. Adiabatic tip
4. Convection (and
combined convection
and radiation).
3-6 Heat Transfer from Finned Surfaces (4)
3-30
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Infinitely Long Fin
/( )cx hp kAmx
b
T x Te e
T T
For a sufficiently long fin the temperature at the fin tip
approaches the ambient temperature
Boundary condition: θ(L→∞)=T(L)-T∞=0
When x→∞ so does emx→∞
C1=0, C2=
@ x=0: emx=1
The temperature distribution:
heat transfer from the entire fin
0
c c b
x
dTQ kA hpkA T T
dx
3-6 Heat Transfer from Finned Surfaces (5)
3-31
b
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Adiabatic Tip
Boundary condition at fin tip:
After some manipulations, the temperature
distribution:
heat transfer from the entire fin
0x L
d
dx
cosh( )
coshb
m L xT x T
T T mL
0
tanhc c b
x
dTQ kA hpkA T T mL
dx
3-6 Heat Transfer from Finned Surfaces (6)
3-32
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
Convection (or Combined Convection and
Radiation) from Fin Tip A practical way of accounting for the heat loss from the fin tip is to replace
the fin length L in the relation for the insulated tip case by a corrected
length defined as
Lc=L+Ac/p
For rectangular and cylindrical
fins Lc is
Lc,rectangular=L+t/2
Lc,cylindrical =L+D/4
3-6 Heat Transfer from Finned Surfaces (7)
3-33
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-7 Bioheat Equation (1)
outin EE ,
Bone (core)
Muscle
Fat
Inner skin
Outer skin
Surroundings (convection, radiation)
Conduction &
heat generation (metabolic, perfusion, etc)
stg EE ,
No heat source
3-34
Ref: Fiala, D., Lomas, K., Stohrer, M., “A Computer Model of Human
Thermogulation for a Wide Range of Environmental Conditions: the
Passive System,” Journal of Physiology, Vol. 87, pp. 1957-1972, 1999.
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
1-D, Steady
)(
02
2
TTcq
k
dx
Td
abbp
pm
: metabolic heat generation rate
: perfusion heat generation rate
mq
pq
3-7 Bioheat Equation (2)
3-35
ω: perfusion rate, m3/s
ρb, cb: blood density and specific heat
Ta: arterial temperature
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-7 Bioheat Equation (3)
1-D, transient
Governing Equation:
t
TcTTcWq
r
T
rr
Tk ablblblblm
)( ,2
2
Initial condition: t=0
Boundary conditions: at surface
3-36
Chapter 3: Steady Heat Conduction Heat Transfer
Y.C. Shih September 2013
3-7 Bioheat Equation (4)
Boundary Conditions
Convection with ambient air, qc: Newton’s cooling law
Radiation with surrounding surfaces, qR
Irradiation from high-temperature sources, qsR
Evaporation of moisture from the skin, qe
qsk
qc qR qsR qe
Skin
Surroundings
Tw T∞ Ein=Eout
qsk+qsR=qc+qR+qe
esRRcsk qqqqq
3-37