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8/10/2019 ME3122 Handbook of Heat Transfer Equations 2014
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HANDBOOK OF EQUATIONS, TABLES
AND CHARTS FOR
ME3122/ME3122E HEAT TRANSFER
Department of Mechanical Engineering
National University of Singapore
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CONDUCTION HEAT TRANSFER
1stlaw of thermodynamics: WQdU
Conduction:
Convection:
Radiation: where -4-28 KWm10675 .
Control Volume:
Surface:
Heat Conduction Equation:
Cartesian:
Cylindrical:
Spherical:
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One-Dimensional Walls
Fin Equations:
kA/hPmmdx
d 22
2
2
where0
which has the general solution mxmx eCeC 21 .
Fin Efficiency:
Fin Effectiveness:
Overall Surface Efficiency:
ftf
t
t
max
t
o A
NA
hA
q
q
q
11
0
whereunfinnedft
ANAA .
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Lumped Capacitance Method:
,
, , ,
Other Equations (Thermal Properties):
Solids:
Free electrons:
Gases:
Joule heating: RIEg2
Interfaces:
Heat wave speed:
Two semi-infinite solids touch:
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CONVECTION HEAT TRANSFER
All symbols have their usual meaning.
ConstantsGravitational acceleration: g = 9.81 m/s2
Specific gas constant for air: R= 287 J/kgK
Definitions
Kinematic viscosity, /
Thermal diffusivity, / pck
Volumetric thermal expansion coefficient,TT p
11
for an ideal gas.
General
Dimensionless Groups
uc
h
PrRe
NuSt
PrGrRa
LTTgGr
k/hLNu
/Pr
/VL/VLRe
p
x
x
xx
LL
sL
L
L
Number,Stanton
Number,Rayleigh
Number,Grashof
Number,Nusselt
Number,Prandtl
Number,Reynolds
2
3
Tcm
y
u
VAm
RTpv
TThq
p
c
s
sectionaghflux throuenergyThermal
stress,Shear
rate,flowMass
:lawgasIdeal
Cooling,ofLawsNewton'
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2D Continuity Equation:
2D x-Momentum Equation:
2D Energy Equation:
where viscous dissipation,
2D Boundary Layer Equations:
x-Momentum Equation:
Energy Equation:
Integral Momentum Equation:
Integral Energy Equation:
Forced Convection Over External Surfaces
Generally,nm
PrReCNu
Forced Convection Over a Flat Plate:
For constant , .
0
y
v
x
u
Xy
u
x
u
x
p
y
u
vx
u
u
2
2
2
2
qy
T
x
Tk
y
Tv
x
Tucp
2
2
2
2
222
2y
v
x
u
x
v
y
u
2
2
y
u
y
uv
x
uu
2
2
y
T
y
Tv
x
Tu
00
)(
yy
udyuuu
dx
d
0
0
yy
TdyTTu
dx
d t
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Uniform Surface Temperature (Isothermal):
11
0
L
x
A
x dxhL
dAhA
h
For laminar flow (5Re 5 10
x ):
;5 3121 PrRex tx
For turbulent flow (5
Re 5 10x ):
Pr02960;05920;370 31
51 5451
xxxx,fxturb Re.NuRe.CRex.
For mixed boundary layer conditions ( 5105LRe ):
Uniform Surface Heat Flux (Isoflux):
For laminar flow (5Re 5 10
x ):
For turbulent flow ( 5Re 5 10x ):3
1
Pr03080 54xx Re.Nu
For Unheated Starting Length,xo, with laminar flow for both isothermal and isoflux
conditions:
Forced Convection Across Long Cylinders:
where Cand mare given byReD C m
0.4-4 0.989 0.330
4-40 0.911 0.385
40-4000 0.683 0.466
4000-40,000 0.193 0.61840,000-400,000 0.027 0.805
31
21
31
21
6640;3320 PrRe.k
LhNuPrRe.Nu LLxx
)8710370(;1742074080151 3
1
.LLLLL,f Re.Prk
LhNuReRe.C
31
21
4530 PrRe.Nu xx
21
21
3281;66402
2
xL,fx
x,s
x,f Re.CRe./u
C
31430
1
x/xNuNu oxxx o
31PrReC
k
DhNu
m
DD
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Forced Convection Across Spheres:
where all properties are evaluated at thefree-streamtemperature, except s, which isevaluated at the surface temperature of the sphere.
Forced Convection Across Non-Circular Cylinders
where Cand mare given by
Forced Convection Across Tube Banks
where all properties, exceptPrs, are evaluated at the average of the fluid inlet and outlet
temperatures,ReD,maxis based on the maximum fluid velocity, and C1and mare given in the
table below for number of tube rows for various aligned and staggered arrangements
of tubes.
41
403221 060402
s
.
DDD
PrRe.Re.
k
DhNu
31PrReC
k
DhNu
m
DD
41
360
1
s
.m
max,DDPr
PrPrReCNu
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(a)Aligned tube rows (b) Staggered tube rows
For : where C2for various is given in the table
below:
20220 LL NDND NuCNu
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Forced Convection in Tubes and Ducts
Friction factor,
PerimeterWettedAreasectional-Cross4Diameter,Hydraulic hD
For thermally fully-developed condition:
Laminar Flow (ReD2300):
Fully developed velocity profile:
where mean fluid velocity,
Friction factor, f= 64/ReD
Nuand ffor Fully-Developed Laminar Flow in Tubes of Various Cross-Sections
dx
dpr
r
mum
8
20
2
0
2
0
2
12)(
r
r
u
ru
m
2
or2
2
2
m
m
u
D
Lfp
/u
Ddx/dpf
0)()(
)()(
xTxT
x,rTxT
x ms
s
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Turbulent Flow (ReD> 2300):
For smooth tubes and ducts, the Dittus-Boelter equation:
with n= 0.4 for heating of fluid, and n= 0.3 for cooling of fluid
Friction factor for smooth tubes: 26417900 .Reln.f D
Friction factor for rough tubes of roughness e: 290745733251 .DRe/.D./eln.f
Reynolds-Colburn Analogy
For flow over a flat plate:
For flow in a tube or duct:
FREE CONVECTION
Generally,
flow.entfor turbul31andflow,laminarfor41with mmRaCPrGrCNu mLm
LL
Laminar Free Convection on an Isothermal Vertical Plate:
Boundary layer momentum equation:
Integral Momentum Equation for Free Convection BL:
Boundary layer thickness,
Critical Ra= 109.
Free Convection from an Isothermal Sphere
n
DD PrRe.Nu hh540230
2;2 3232 /CPr.St/CPr.St L,fLx,fx
832 /fPr.St
2
2
y
uTTg
y
uv
x
uu
00
2dyTTg
y
udyu
dx
d
s
414121 9520933 xGrPr.Prx.
541 101for4302 D/
DD GrPrGr.k
DhNu
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Free Convection from Isothermal Planes and Cylinders
Free Convection from a Vertical Plate with Constant Surface Heat Flux
where
mLm
LL RaCPrGrCNu
Geometry GrL Pr C mCharacteristic
Length
Vertical plane and cylinder104109 0.59 1/4
Height10 10 0.10 1/3
Horizontal cylinder
10-1010-2 0.68 0.058
Diameter
10- 10 1.02 0.148
102104 0.85 0.188
104109 0.53 1/4
1091012 0.13 1/3
Hot surface facing up or
cold surface facing down
104107 0.54 1/4Area/Perimeter
1071011 0.15 1/3
Hot surface facing down or
cold surface facing up1051011 0.27 1/4 Area/Perimeter
161341
11551
10102for170:Turbulent
1010for600:Laminar
Pr*GrPr*Gr.Nu
Pr*GrPr*.Gr.k
xhNu
xxx
xxx
x
2
4
k
xqg.NuGr*Gr sxxx
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RADIATION HEAT TRANSFER
Solid angle: 2/ rAn , ddsind
Radiation:where
-4-28
KWm10675
.
surssursr
sursr
"
rad
TTTTh
TThq
22
Spectral directional Intensity:
Diffuse emitter:
Blackbody: 4)( TTEb
Spectral black body emissive power
).)T/Cexp(
C)T,(E b, m(W/m
1
2
2
5
1
m.K104391and/mmW.107423where 42248
1 .C.C
Weins displacement law: m.K2898max T
Emissivity of real surfaces:
4)()()( TTETTE b
Absorptivity of surface:
GGabs
Semitransparent medium: 1
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Black Body Radiation Functions
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View factors:
32
133122132
AA
FAFAF ,
Radiation exchange between black-body surfaces:
Radiation network approach:
resistancespatial1re whe1
resistancesurface1where1
121
121
2112
FA/FA/
JJq
A/A/
JEq b
Radiation Exchange Network for a Two-Surface Enclosure
22
2
21111
1
4
2
4
112
111AFAA
TTq
,
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View factor for aligned parallel rectangles
View factor for coaxial parallel disks
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View factor for perpendicular rectangles with common edge
HEAT EXCHANGERS
Log Mean Temperature Difference, io
iolm
T/Tln
TTT
where fiR and fiR are fouling factors.
Capacity rate, pcmC , is infinite for a condensing or boiling fluid.
ooo
fo
w
i
fi
ii
BA
AhA
RR
A
R
Ah
TTq
11
exchangerheatindifferenceatureMax temper
fluid)(minimum
rateferheat transpossibleMax
rateferheat transActualess,Effectiven
T
max
min
max
minRatio,RateCapacityC
C
cm
cmCr
nits)Transfer Uof(NumberNTUC/UA min
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Correction Factor for Single Pass Cross-Flow Heat
Exchangers with the Shell Side Fluid Mixed, and the
Other Fluid Unmixed.
Correction Factor for a Single Pass Cross-Flow
Heat Exchanger withBoth Fluids Unmixed.
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-NTU Charts for Heat Exchangers
Effectiveness of parallel flow heat exchangers Effectiveness of counterflow heat exchangers
Effectiveness of Heat Exchangers with One Shell
Passand Two (or Multiples of Two)Tube Passes.
Effectiveness of Heat Exchangers with Two Shell
PassesandFour (or Multiples of Four)Tube Passes.
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Heat Exchanger Effectiveness Relations
Heat Exchanger NTU Relations
Use the above two equations with