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MECH-346 Heat Transfer: Summary Sheets Complete Version 1
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MECH-346 Heat Transfer: Summary Sheets Complete Version
Fouriers law of heat conduction for isotropic materials:
"q k T=
Convection heat loss from an isothermal surface:
Net radiation loss from a flat isothermal surface to surroundings atsurr
T
(or a large enclosure with isothermal walls atw surr
T T= ): assumption is
that surface is gray w.r.t. radiation from surroundings, sosurf surf
=
Three-dimensional unsteady heat conduction in an isotropic material:
General (vector calculus) form of the governing equation:
( ). pT
k T S ct
+ =
Asur ace
Tsur
T
hav
, ( )conv loss surface av surf exposedtoconv
q A h T T =
,( )conv surf th convq T T R
( ), 1th conv surface exposed t o conv avR A h
surr wT or T
, ,surf surf surf T A
4 4
.
net surf surf surf surr radiation abs abssurface surrorsurface encl walls
q A T T
=
8
2 4
Stefan-Boltzmann constant
5.669x10 [ ]W
m K
=
=
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Governing equation expressed in Cartesian coordinates:
p
T T T T k k k S c
x x y y z z t
+ + + =
Governing equation expressed in cylindrical coordinates:
2
1 1p
T T T T rk k k S c
r r r r z z t
+ + + =
Governing equation expressed in spherical coordinates:
2
2 2 2 2
1 1 1(sin )
sin sinp
T T T Tr k k k S c
r r r r r t
+ + + =
Steady-State One-Dimensional Heat Conduction [Isotropic Materials]
1. Plane Wall
T1
T2
L
x
Ac.s.
Governing equation (1-D Cartesian):
0d dT
k Sdx dx
+ =
For constantk= and 0S= , the solution is:
2 1 2( ) ( ) 1 ( / )T T T T x L =
. . 1 2 1 2. .
, //
, //. .
( ) ( )( )
Restrictions: 1-D, steady-sta
0, ., // wall
c sx c s
th wall
th wallc s
A k T T T TdTq k A
dx L R
L
R S k const kA
= =
=
= =
1T 2T xq
, //th wallR
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T1 T2
qr
Rth,long hollow cylinder
Plane wall of thickness 2L: S.S., 1-D Cartesian, S= constant, k= constant
2. Long Hollow Cylinder [L>> (r2 r1)]
Resistance analogy: Long hollow cylinder ( 0r ), steady-state, 1-D radialheat conduction, k= constant, and 0S= are the restrictions here
L
x
Ac.s.L
S>0(const.)
Tx=L
= TRight
General case: (i) atx= -L, T= TLeftand
(ii) atx=L, T= TRight
22
12 2 2
Right Left Right LeftT T T T S L x xT
k L L
+ = + +
Symmetric case: (i) atx= -L, T= TWand
(ii) atx=L, T= TW
22 2
max 0
1 ;2 2W x W
S L x S LT T T T T
k L k=
= + = = +
T1
T2
r2r1
r
r
L
Steady-state, 1-D radial heat conduction: Governing equation
Tx=-L
= TLeft
1 0d dTrk Sr dr dr
+ =
k = constant, S = 0, with B.C.s;(i) at 1 1,r r T T = = ; and (ii) at 2 2,r r T T= =
Solution for this case is the following:
2 1 2 2 1 2( ) /( ) ln( / ) / ln( / )T T T T r r r r =
2 1,
ln( / )
2th longhollow cylinder
r rR
kL=
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T1 T2
qr
Rth,hollow sphere
3. Long solid cylinder (L>> R): steady-state, 1-D radial heat conduction
4. Hollow Sphere
Resistance analogy: Hollow sphere ( 0r ), steady-state, 1-D radial heat
conduction, k= constant, and 0S= are the restrictions here
r2r1
Tw
R
L
Governing equation:
10
d dTrk S
r dr dr
+ =
k= constant, S= constant, with B.C.s;(i) at 0, is finiter T= ; and (ii) at , wr R T T = =
Solution for this case is the following:22
( ) 14
w
S R rT T
k R
=
; and
2
max 04
r wS RT T T
k== = +
Steady-state, 1-D radial heat conduction: Governing equation
2
2
10
d dTr k S
r dr dr
+ =
k= constant, S= 0, with B.C.s;(i) at 1 1,r r T T = = ; and (ii) at 2 2,r r T T= =
Solution for this case is the following:
2
2 1 21 2
( ) 1 1 1 1
( )
T T
r r r r T T
=
,
1 2
1 1 1
4th hollowsphereR
k r r
=
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5. Solid sphere: steady-state, 1-D radial heat conduction
Thermal Contact Resistance
( ) ( ),th contact contact interfaceI I I Iq T T R h A T T + +
Thus, ( ), 1th contact contact interfaceR h A=
Here, contacth is the thermal contact coefficient [ 2 oW
m C]
R
Governing equation:
2
2
10
d dTr k S
r dr dr
+ =
k= constant, S= constant, with B.C.s;(i) at 0, is finiter T= ; and (ii) at , wr R T T = =
Solution for this case is the following:22
( ) 16
w
S R rT T
k R
=
; and
2
max 06
r wS RT T T
k== = +
Tw
Interface
Material Material I
T IT +
q
Note: For unit
contact area, thethermal contact
resistance is
denoted as:"
, 1/th contact contaR h=
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Critical Radius of Insulation or Coating of Curved Surfaces
(Conduction-Convection Systems)
Long hollow cylindrical geometryIf { }1 1, , , , ,inslr L k T T h all constant, 2r variable; 0inslS = ; steady-state; and
1-D radial, then when
2insl
critlong hollow cyl
kr r
h= = , maxq q=
Hollow spherical geometry
If { }1 1, , , , ,inslr L k T T h all constant, 2r variable; 0inslS = ; steady-state; and
1-D radial, then when
2
2insl
crithollowsphere
kr r
h= = , maxq q=
Note:For both the cylindrical and spherical geometries, if
{ }1, , , , ,inslr L k h T q all constant, 2r variable; 0inslS = ; steady-state; and 1-D
radial, then when 2 critr r= , 1 1( )minimumT T=
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Classical Fin Theory
(fink = constant; finS =0; h= constant; T = constant; steady-state)
For uniform fin cross-sectional area and perimeter, . .c sA = constant and
. .c sPeri = constant, and the fin equation reduces to2
1/ 22
. . . .2( ) 0 ; ( ) /( )
c s fin c s
d Tm T T m h Peri k A
dx
= =
B.C. (i): atx= 0, 0x baseT T= =
For B.C. (ii), the following four cases and solutions are considered:
Case A:at x L= ,
}( / ) ( ) Convection heat loss from the tip surfacefin x L x Lk dT dx h T T = = =
Case A solution:
[ ]cosh ( ) sinh[ ( )]
cosh( ) sinh( )
fin
base
fin
hm L x m L x
m kT T
T T hmL mL
m k
+ =
+
Case A total rate of heat transfer from fin to fluid:
( ). . . .sinh( ) cosh( )
( )
cosh( ) sinh( )
fin
total fin c s c s basefin fluid
fin
hmL mLm k
q k A h Peri T T h
mL mLm k
+
= +
Case B:at x L= ,
}( / ) 0 Adiabatic tip or symmetry surface atx LdT dx x L= = =
Case B solution: [ ]cosh ( )
cosh( )base
m L xT T
T T mL
=
Case B total rate of heat transfer from fin to fluid:
( ). . . . ( ) tanh( )total fin c s c s basefin fluid
q k A h Peri T T mL
=
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Case C:Long-fin, at x L= ,x L
T T= =
Case C solution: ( )expbase
T Tmx
T T
=
Case B total rate of heat transfer from fin to fluid:( ). . . . ( )total fin c s c s base
fin fluid
q k A h Peri T T
=
Case D:at x L= , } Specified fin tip temperaturex L LT T= =
Case D solution:
sinh( ) sinh[ ( )]
sinh( )
L
base
base
T Tmx m L x
T TT T
T T mL
+ =
Case D total rate of heat transfer into fin across base:
( ). . . .cosh( )
( )sinh( )
L
base
total fin c s c s basebase in
T TmL
T Tq k A h Peri T T
mL
=
Case D total rate of heat transfer from lateral surface of fin to fluid:
. .lateral surface total out total fin c sfin fluid base in fin tip base in tipx L
dTq q q q k Adx =
= = +
Fin Efficiency:,
Entire fin at
Same fin geometry and tip condition
base
actual fin fluid
fin T
fin fluid
q
q
For Case B (insulated tip): ( ), tanhfin Case B mL mL =
Compensated length:c
L may be used to approximate the total rate of heat
loss from the fin to the fluid in Case A using the results for Case B: in general,
( ). . . .c c s tip c s tipL L L L A Peri= + + ; for a straight fin of uniformrectangular cross-section with W >> t, / 2cL L t= + ; for a straight fin ofuniform circular cross-section, with diameter d, / 4cL L d= +
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Fin Thermal Resistance: , ,1 [ ]th fin fin surface total fin fluid R Area h ; note that this
fin thermal resistance accounts for both conduction through the fin and
convection at its surface
Fin Effectiveness:,
, ( )
actual fin fluid
fin
base fin w
q
Area h T T
Fin Design Charts and Related Procedures
Basis: Solutions based on adiabatic fin tip assumption (akin to Case B
solution for a fin of uniform rectangular cross-section)
Thus, when using fin design charts, use Lc instead of L if the fin tip
looses heat by convection (but if the actual fin tip is adiabatic, then use
the actual fin length,L).
3/ 2 1/ 2[ { /( )} , ]fin c fin m
fnc L h k A geometric paramters = ;Amis the profile area
If the fin efficiency,fin
, is obtained from fin design charts, then the fin
thermal resistance is ,,
,
1/( )c
th fin fin surface lateral totalfin fluid withL if needed
R Area h=
If
fin is obtained from fin design charts, then use the following
expressions:,
,,
( )( )
c
base
actual total fin surface lateral basefin fluid total fin
th finwith L if needed
T Tq Area h T T
R
= =
Design charts for uniform fins of triangular cross-section, uniform fins
of rectangular cross-section, and circumferential (or annular) fins of
rectangular cross-section are given on the following page: again, use the
compensated fin length,Lc, if needed (that is, if the fin tip looses heat by
convection); but if the actual fin tip is adiabatic, then use the actual fin
length,L.
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Fin efficiency charts: Fin
of uniform rectangular
and triangular cross-
sections
rectangular fi2
triangular fin
rectangular
triangular fi2
c
c
m
c
tL
L
L
tL
A tL
+
=
=
Figures taken from Heat Transfer by J.P. Holman, 7th
Edition, 1990 Fin efficiency
chart:
Circumferential or
annular fins ofuniform
rectangular cross-
section
2 1
2 1
2
( )
c
c c
m c
tL L
r r L
A t r r
= +
= +
=
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Heat Conduction Shape Factor
A heat conduction shape factor, S, can be defined in problems that allow the
following approximations or restrictions: isotropic and homogeneous material;
steady-state conditions prevail; k= constant; volumetric source term S = 0; and
only two different uniform boundary temperatures.
1 2
1 2
1 2
( )( )
totalT T
th cond
T Tq k T T
R
= =S where 1/( )
th condR k= S
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Unsteady Heat Conduction (Isotropic Materials)
Governing equation: ( ) /
pdiv k T S c T t + =
Properties and source data ( , , ,p
k c S ), B.C.s ,and I.C. are needed to
complete the mathematical model: these are problem specific.
Biot number (conduction-convection systems):
c
solid
hLBi
k ; where
[Total volume of solid]
[Surface area of solid exposed to convection]c
L
Lumped Parameter Analysis [LPA; valid if 0.1Bi ]
Key idea
Governing equation: ( ) ( / )solid surface solid p solid
exposed to convection
SV A h T T c V dT dt =
where Sis the volumetric source term
LPA solution for 0S= , [ , , , ]p
c h T all constant,0t ini
T T uniform= = = :
( )exp
( )
surface exposedto convection
ini ini p solid
h AT T
tT T c Vol
= =
Solid
Fluid, T, h( , , , ) ( )T T x y z t T t =
when
0.1c
solid
h LBi
k
; with
( )solidc
surface solid exposed to convection
VolumeL
Area=
( )p solid surface exposedto convection
c Vol h A
=
time constant of conduction-convection
system
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When LPA solution for 0S= , [ , , , ]pc h T all constant, and
0t iniT T uniform= = = applies; and the time constant of the conduction-convec
system is ( )p solid surface solid exposedto convection
c Vol h A =
:
1 2
1 2
1 2( ) ( ) exp exptotal loss solid p t t t t solid p init t t
t tQ Vol c T T Vol c T T
= =
= =
Transient heat conduction in semi-infinite solids
Mathematical model:with the thermal diffusivity /( )p
k c =
Governing equation:p
T Tk c
x x t
=
or2
2
1T T
x t
=
x
k= const.; S= 0;
= const.; cp= const.
uniformi
T T= = ,for 0t
Isotropic materials
Common B.C.s
(imposed) on the
surface: for 0t> a)Constant temperature,
o iT T
b)Constant heat flux,0q
c)Convection boundary
condition, ,h T
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I.C.: at t= 0, T= Ti= constant; 0 x
B.C.s
a)0, .
0, .
o
i
at x T T const for all t
at x T T const
= = = > = =
b) 00, .
0
, .
o
x
i
Tat x q k const
x for all t
at x T T const
=
= = =
> = =
c)0
0
0, ( )
0, .
x
x
i
Tat x h T T k
x for all t
at x T T const
=
=
= =
> = =
Solutions:
a)( , )
erf2
o
i o
T x t T x
T T t
=
Notes:
1) Error function: ( )2
0
2erf e d
; values in Table 8-1 (attached)
2) Complementary error function: ( ) ( )erfc 1 erf
b)22 / -
( , ) exp - erfc4 2
o oi
q t q xx xT x t T
k t k t
=
c)2
2
( , )erfc exp erfc
2 2
i
i
T x t T x hx h t x h t
T T k k k t t
= + +
Penetration depth: ( )t
When ( )x t= ,
0( , ) 0.992i o
T t Terf
T T t
= =
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Table 8.1: Values of error function
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One-dimensional unsteady heat conduction in solids with
convection B.C. [S =0; k, , cp, h, T all constant; cLBi > 0.1;
uniform initial temperature,i
T; and /( )pk c = ]
Case B:
L>> ro
ro
h, T
Long solid cylinder
oM
solid
hrBi
k= ; *
2o
tt
r
= ; *
o
rr
r=
Case A:
T
h
T
h
LL
t 0, T=Ti
Symmetrically cooled/heated
plane wall
M
solid
hLBi
k= ; *
2
tt
L
= ; *
xx
L=
Case C:
ro
h, T
Solid sphere
oM
solid
hrBi
k= ; *
2o
tt
r
= ; *
o
rr
r=
Notes:
1.One-dimensional
transient heat conduction
in these three cases can
be predicted analytically:
solutions are in the form
of infinite series;
2.
However, these series
are rapidly convergent;
3.For * 0.2t , one-termapproximation of infinite
series is excellent:
[Error 2%]
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For * 0.2t
, use the following one-term approximation:
Case A:Symmetrically cooled/heated plane wall (total thickness = 2L)
* * * 2 * * * * 21 1 1
( , )( , ) exp[ ]cos( ) ; / ; /
i
T x t T x t A t x x x L t t L
T T
= = = =
* * * 2 *1 1
( 0, )( 0, ) exp[ ]
i
T x t T x t A t
T T
= = = =
*
* * * 1
, 101
sin( )1 [ ( 0, )] ; 2 ; ( )
t
o total p iloss xo
Qx t Q q dt Q Vol c T T
Q
=
= = = =
Case B:Long solid cylinder (radiuso
r ; cylinder length >> 2or)
* * * 2 * * * * 21 1 0 1
( , )( , ) exp[ ]J ( ) ; / ; /
o o
i
T r t T r t A t r r r r t t r
T T
= = = =
* * * 2 *1 1
( 0, )( 0, ) exp[ ]
i
T r t T r t A t
T T
= = = =
*
* * * 1 1
, 101
J ( )1 2[ ( 0, )] ; ; ( )
t
o total p iloss ro
Qr t Q q dt Q Vol c T T
Q
=
= = = =
Case C:Solid sphere (radiusor)
** * * 2 * * * 21
1 1 *
1
( , ) sin( )( , ) exp[ ] ; / ; /
o o
i
T r t T r r t A t r r r t t r
T T r
= = = =
* * * 2 *1 1
( 0, )( 0, ) exp[ ]
i
T r t T r t A t
T T
= = = =
*
* * * 1 1 13 , 10
1
sin( ) cos( )1 3[ ( 0, )] ; ; (
t
o total p iloss ro
Qr t Q q dt Q Vol c T T
Q
=
= = = =
Notes:
For cases A, B, and C, values of 1 1,A as functions of MBi are givenin Table 9.1 (see page 18)
Values of 0 1J ( ) and J ( ) as functions of are given in Table 9.2
(see page 18). Note: o 1J ( ) / J ( )d d =
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Table 9.1 Table 9.2
Values of 1and 1A for different values of Zeroth- and
MBi and Cases A, B, and C first-order
Bessel functionsof the first kind
Notes:
( ) ; ( ) ; ( )o oM PlaneWall M Long Solid Cylinder M Solid Sphere
solid solid solid
hr hr hLBi Bi Bi
k k k= = =
( / 2) ( /3)( ) ; ( ) ; ( )
c c c
o oL PlaneWall L Long Solid Cylinder L Solid Sphere
solid solid solid
h r h r hLBi Bi Bi
k k k= = =
Case A Case B Case C
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Product solution approach: temperature distribution for
multidimensional unsteady heat conduction in solids with convection B.C.
[S=0; k, , cp, h, T all constant; cLBi > 0.1; and iTuniform]
Notation for one-dimensional unsteady solutions:
Plane-wall: ( , )( , )i
T x t T P x tT T
=
Long-cylinder:( , )
( , )i
T r t T C r t
T T
=
Semi-infinite solid:( , )( , )
( , ) 1 i
i i
T x t T T x t T x t
T T T T
= =
S
In general, for three-dimensional unsteady
problems:3
13 ISi iFull D solid Intersecting So
T T T T
T T T T
=
=
P= S 1 2P P=
1 2 3P P= S 1 2 3P P P=
C= S P C=
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Total heat transfer for multidimensional unsteady heatconduction in solids with convection B.C. [S=0; k, , cp, h, T all
constant; iTuniform; and cLBi > 0.1]
Work of Langston
For solids that can be constructed by the intersection of two objects (1
and 2) for which the 1-D solutions (discussed earlier) apply:
1 2 1
1o o o ototal
Q Q Q Q
Q Q Q Q
= +
For solids that can be constructed by the intersection of three objects
(1, 2, and 3) for which the 1-D solutions (discussed earlier) apply:
1 2 1
3 2 1
1
1 1
o o o ototal
o o o
Q Q Q Q
Q Q Q Q
Q Q Q
Q Q Q
= +
+
If the temperature distribution and/or the total heat transferafter a given time into the heating/cooling process is desired, the
solution is straightforward
Obtain the appropriate one-dimensional solutions and combine
them suitably, as discussed above
If the time needed to obtain a desired temperature distribution or
total heat transfer is required, then:
Explore options offered by symmetry surfaces
Keep your thinking cap on
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Uniform Newtonian Fluid Flow and Heat Transfer over aFlat Plate with a Sharp Leading Edge and Zero Angle of Attack
Local or running Reynolds number, /
xRe U x
Transition region:
5 61 x 10 1 x 10xRe
In engineering analyses:
5( / ) 5 x 10critx crit
Re U x = = for flow over
a flat plate at zero angle of attack
At , ( / ) 0.99y u U = =
At , [( ) /( )] 0.99T w wy T T T T = =
Fully Turbulent Layer:turb >>
Buffer Layer:turb
Viscous sublayer:turb