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hol29362_ifc 10/30/2008 18:42
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.1 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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Useful conversion factors
Physical quantity Symbol SI to English conversion English to SI conversion
Length L 1 m = 3.2808 ft 1 ft = 0.3048 mArea A 1 m2 = 10.7639 ft2 1 ft2 = 0.092903 m2Volume V 1 m3 = 35.3134 ft3 1 ft3 = 0.028317 m3Velocity v 1 m/s = 3.2808 ft/s 1 ft/s = 0.3048 m/sDensity ρ 1 kg/m3 = 0.06243 lbm/ft3 1 lbm/ft3 = 16.018 kg/m3Force F 1 N = 0.2248 lbf 1 lbf = 4.4482 NMass m 1 kg = 2.20462 lbm 1 lbm = 0.45359237 kgPressure p 1 N/m2 = 1.45038 × 10−4 lbf /in2 1 lbf /in2 = 6894.76 N/m2Energy, heat q 1 kJ = 0.94783 Btu 1 Btu = 1.05504 kJHeat flow q 1 W = 3.4121 Btu/h 1 Btu/h = 0.29307 WHeat flux per unit area q/A 1 W/m2 = 0.317 Btu/h · ft2 1 Btu/h · ft2 = 3.154 W/m2Heat flux per unit length q/L 1 W/m = 1.0403 Btu/h · ft 1 Btu/h · ft = 0.9613 W/mHeat generation per unit volume q̇ 1 W/m3 = 0.096623 Btu/h · ft3 1 Btu/h · ft3 = 10.35 W/m3Energy per unit mass q/m 1 kJ/kg = 0.4299 Btu/lbm 1 Btu/lbm = 2.326 kJ/kgSpecific heat c 1 kJ/kg · ◦C = 0.23884 Btu/lbm · ◦F 1 Btu/lbm · ◦F = 4.1869 kJ/kg · ◦CThermal conductivity k 1 W/m · ◦C = 0.5778 Btu/h · ft · ◦F 1 Btu/h · ft · ◦F = 1.7307 W/m · ◦CConvection heat-transfer coefficient h 1 W/m2 · ◦C = 0.1761 Btu/h · ft2 · ◦F 1 Btu/h · ft2 · ◦F = 5.6782 W/m2 · ◦CDynamic 1 kg/m · s = 0.672 lbm/ft · sViscosity μ = 2419.2 lbm/ft · h 1 lbm/ft · s = 1.4881 kg/m · sKinematic viscosity and thermal diffusivity ν, α 1 m2/s = 10.7639 ft2/s 1 ft2/s = 0.092903 m2/s
Important physical constants
Avogadro’s number N0 = 6.022045 × 1026 molecules/kg molUniversal gas constant R= 1545.35 ft · lbf/lbm · mol · ◦R
= 8314.41 J/kg mol · K= 1.986 Btu/lbm · mol · ◦R= 1.986 kcal/kg mol · K
Planck’s constant h= 6.626176 × 10−34 J · secBoltzmann’s constant k= 1.380662 × 10−23 J/molecule · K
= 8.6173 × 10−5 eV/molecule · KSpeed of light in vacuum c= 2.997925 × 108 m/sStandard gravitational acceleration g= 32.174 ft/s2
= 9.80665 m/s2
Electron mass me= 9.1095 × 10−31 kgCharge on the electron e= 1.602189 × 10−19 CStefan-Boltzmann constant σ= 0.1714 × 10−8 Btu/hr · ft2 · R4
= 5.669 × 10−8 W/m2 · K4
1 atm = 14.69595 lbf/in2 = 760 mmHg at 32◦F= 29.92 inHg at 32◦F = 2116.21 lbf/ft2= 1.01325 × 105 N/m2
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hol29362_ifc 10/30/2008 18:42
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.2 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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Basic Heat-Transfer Relations
Fourier’s law of heat conduction:
qx = −kA∂T∂x
Characteristic thermal resistance for conduction =�x/kACharacteristic thermal resistance for convection = 1/hAOverall heat transfer =�Toverall/�Rthermal
Convection heat transfer from a surface:
q=hA(Tsurface − Tfree stream) for exterior flowsq=hA(Tsurface − Tfluid bulk) for flow in channels
Forced convection: Nu = f(Re, Pr) (Chapters 5 and 6, Tables 5-2 and 6-8)Free convection: Nu = f(Gr, Pr) (Chapter 7, Table 7-5)
Re = ρuxμ
Gr = ρ2gβ�Tx3
μ2Pr = cpμ
k
x= characteristic dimensionGeneral procedure for analysis of convection problems: Section 7-14, Figure 7-15, Insideback cover.
Radiation heat transfer (Chapter 8)
Blackbody emissive power,energy emitted by blackbody
area · time = σT4
Radiosity = energy leaving surfacearea · time
Irradiation = energy incident on surfacearea · time
Radiation shape factor Fmn= fraction of energy leaving surface mand arriving at surface n
Reciprocity relation: AmFmn=AnFnmRadiation heat transfer from surface with area A1, emissivity 1, and temperature T1(K) tolarge enclosure at temperature T2(K):
q= σA11(T 41 − T 42 )LMTD method for heat exchangers (Section 10-5):
q=UAF �Tmwhere F = factor for specific heat exchanger; �Tm= LMTD for counterflow double-pipeheat exchanger with same inlet and exit temperatures
Effectiveness-NTU method for heat exchangers (Section 10-6, Table 10-3):
= Temperaure difference for fluid with minimum value of mcLargest temperature difference in heat exchanger
NTU = UACmin
= f(NTU, Cmin/Cmax)See List of Symbols on page xvii for definitions of terms.
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Hol29362_Ch01 10/7/2008 18:49
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.6 K/PMS 293
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6 1-2 Thermal Conductivity
liquids and solids, but in general, many open questions and concepts still need clarificationwhere liquids and solids are concerned.
The mechanism of thermal conduction in a gas is a simple one. We identify the kineticenergy of a molecule with its temperature; thus, in a high-temperature region, the moleculeshave higher velocities than in some lower-temperature region. The molecules are in contin-uous random motion, colliding with one another and exchanging energy and momentum.The molecules have this random motion whether or not a temperature gradient exists in thegas. If a molecule moves from a high-temperature region to a region of lower temperature,it transports kinetic energy to the lower-temperature part of the system and gives up thisenergy through collisions with lower-energy molecules.
Table 1-1 lists typical values of the thermal conductivities for several materials toindicate the relative orders of magnitude to be expected in practice. More complete tabularinformation is given in Appendix A. In general, the thermal conductivity is stronglytemperature-dependent.
Table 1-1 Thermal conductivity of various materials at 0◦C.Thermal conductivity
k
Material W/m · ◦C Btu/h · ft · ◦FMetals:
Silver (pure) 410 237Copper (pure) 385 223Aluminum (pure) 202 117Nickel (pure) 93 54Iron (pure) 73 42Carbon steel, 1% C 43 25Lead (pure) 35 20.3Chrome-nickel steel (18% Cr, 8% Ni) 16.3 9.4
Nonmetallic solids:Diamond 2300 1329Quartz, parallel to axis 41.6 24Magnesite 4.15 2.4Marble 2.08−2.94 1.2−1.7Sandstone 1.83 1.06Glass, window 0.78 0.45Maple or oak 0.17 0.096Hard rubber 0.15 0.087Polyvinyl chloride 0.09 0.052Styrofoam 0.033 0.019Sawdust 0.059 0.034Glass wool 0.038 0.022Ice 2.22 1.28
Liquids:Mercury 8.21 4.74Water 0.556 0.327Ammonia 0.540 0.312Lubricating oil, SAE 50 0.147 0.085Freon 12, CCl2F2 0.073 0.042
Gases:Hydrogen 0.175 0.101Helium 0.141 0.081Air 0.024 0.0139Water vapor (saturated) 0.0206 0.0119Carbon dioxide 0.0146 0.00844
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Hol29362_Ch01 10/7/2008 18:49
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.11 K/PMS 293
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C H A P T E R 1 Introduction 11
thermal properties of the fluid (thermal conductivity, specific heat, density). This is expectedbecause viscosity influences the velocity profile and, correspondingly, the energy-transferrate in the region near the wall.
If a heated plate were exposed to ambient room air without an external source of motion,a movement of the air would be experienced as a result of the density gradients near theplate. We call this natural, or free, convection as opposed to forced convection, whichis experienced in the case of the fan blowing air over a plate. Boiling and condensationphenomena are also grouped under the general subject of convection heat transfer. Theapproximate ranges of convection heat-transfer coefficients are indicated in Table 1-3.
Convection Energy Balance on a Flow Channel
The energy transfer expressed by Equation (1-8) is used for evaluating the convection lossfor flow over an external surface. Of equal importance is the convection gain or loss resultingfrom a fluid flowing inside a channel or tube as shown in Figure 1-8. In this case, the heatedwall at Tw loses heat to the cooler fluid, which consequently rises in temperature as it flows
Table 1-3 Approximate values of convection heat-transfer coefficients.
h
Mode W/m2 · ◦C Btu/h · ft2 · ◦FAcross 2.5-cm air gap evacuated to a pressure
of 10−6 atm and subjectedto �T = 100◦C − 30◦C 0.087 0.015
Free convection, �T = 30◦CVertical plate 0.3 m [1 ft] high in air 4.5 0.79Horizontal cylinder, 5-cm diameter, in air 6.5 1.14Horizontal cylinder, 2-cm diameter,
in water 890 157Heat transfer across 1.5-cm vertical air
gap with �T = 60◦C 2.64 0.46Fine wire in air, d= 0.02 mm,�T = 55◦C 490 86
Forced convectionAirflow at 2 m/s over 0.2-m square plate 12 2.1Airflow at 35 m/s over 0.75-m square plate 75 13.2Airflow at Mach number = 3, p= 1/20 atm,T∞ = −40◦C, across 0.2-m square plate 56 9.9
Air at 2 atm flowing in 2.5-cm-diametertube at 10 m/s 65 11.4
Water at 0.5 kg/s flowing in 2.5-cm-diametertube 3500 616
Airflow across 5-cm-diameter cylinderwith velocity of 50 m/s 180 32
Liquid bismuth at 4.5 kg/s and 420◦Cin 5.0-cm-diameter tube 3410 600
Airflow at 50 m/s across fine wire,d= 0.04 mm 3850 678
Boiling waterIn a pool or container 2500–35,000 440–6200Flowing in a tube 5000–100,000 880–17,600
Condensation of water vapor, 1 atmVertical surfaces 4000–11,300 700–2000Outside horizontal tubes 9500–25,000 1700–4400
Dropwise condensation 170,000–290,000 30,000–50,000
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hol29362_ch02 10/13/2008 18:58
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.30 K/PMS 293
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30 2-4 Radial Systems
Table 2-1 Insulation types and applications.
ThermalTemperature conductivity, Density,
Type range, ◦C mW/m · ◦C kg/m3 Application1 Linde evacuated superinsulation −240–1100 0.0015–0.72 Variable Many2 Urethane foam −180–150 16–20 25–48 Hot and cold pipes3 Urethane foam −170–110 16–20 32 Tanks4 Cellular glass blocks −200–200 29–108 110–150 Tanks and pipes5 Fiberglass blanket for wrapping −80–290 22–78 10–50 Pipe and pipe fittings6 Fiberglass blankets −170–230 25–86 10–50 Tanks and equipment7 Fiberglass preformed shapes −50–230 32–55 10–50 Piping8 Elastomeric sheets −40–100 36–39 70–100 Tanks9 Fiberglass mats 60–370 30–55 10–50 Pipe and pipe fittings
10 Elastomeric preformed shapes −40–100 36–39 70–100 Pipe and fittings11 Fiberglass with vapor −5–70 29–45 10–32 Refrigeration lines
barrier blanket12 Fiberglass without vapor to 250 29–45 24–48 Hot piping
barrier jacket13 Fiberglass boards 20–450 33–52 25–100 Boilers, tanks,
heat exchangers14 Cellular glass blocks and boards 20–500 29–108 110–150 Hot piping15 Urethane foam blocks and 100–150 16–20 25–65 Piping
boards16 Mineral fiber preformed shapes to 650 35–91 125–160 Hot piping17 Mineral fiber blankets to 750 37–81 125 Hot piping18 Mineral wool blocks 450–1000 52–130 175–290 Hot piping19 Calcium silicate blocks, boards 230–1000 32–85 100–160 Hot piping, boilers,
chimney linings20 Mineral fiber blocks to 1100 52–130 210 Boilers and tanks
Figure 2-3 One-dimensional heat flowthrough a hollow cylinderand electrical analog.
L
q
rri
ro dr
Ti
R th =ln(ro/ri)2 kL π
q
To
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50 2-10 Fins
Figure 2-11 Efficiencies of straight rectangular and triangular fins.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5
L
tLc �
L � 2 rectangular L triangular
Am � tLc rectangular
2 Lc triangulart
t
L
t
Lc3/2(h/kAm)
1/2
Fin
effi
cien
cy,
ƒη
Figure 2-12 Efficiencies of circumferential fins of rectangularprofile, according to Reference 3.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.5 1 1.5 2 2.5 3
t
L r1r2
Lc3�2(h�kAm)1�2
Lc = L +
r2c = r1 + Lc
L = r2 − r1
Am = tLc
Fin
effi
cien
cy,
fη
t2
1
2
34 5
r2c�r1
number of applications, and the interested reader should consult Siegel and Howell [9] forinformation on this subject.
In some cases a valid method of evaluating fin performance is to compare the heattransfer with the fin to that which would be obtained without the fin. The ratio of thesequantities is
q with fin
q without fin= ηfAfhθ0
hAbθ0
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hol29362_ch03 10/29/2008 15:34
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.84 K/PMS 293
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84 3-4 The Conduction Shape Factor
Table 3-1 Conduction shape factors, summarized from References 6 and 7.Note: For buried objects the temperature difference is �T = Tobject − Tfar field. The far-fieldtemperature is taken the same as the isothermal surface temperature for semi-infinite media.
Physical system Schematic Shape factor Restrictions
Isothermal cylinder of radius rburied in semi-infinite mediumhaving isothermal surface
D
r
L
Isothermal2πL
cosh−1(D/r)2πL
ln(D/r)
L� r
L� rD> 3r
Isothermal sphere of radius rburied in infinite medium
r
4πr
Isothermal sphere of radius rburied in semi-infinite mediumhaving isothermal surface�T = Tsurf − Tfar field D
r
Isothermal 4πr
1 − r/2D
Conduction between twoisothermal cylinders of lengthLburied in infinite medium
D
r1 r2 2πL
cosh−1(D2−r21−r22
2r1r2
) L� rL�D
Row of horizontal cylindersof length L in semi-infinitemedium with isothermalsurface
Isothermal
D
r
l
S= 2πLln[(
1πr
)sinh(2πD/l)
] D> 2r
Buried cube in infinitemedium, L on a side
L
8.24L
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C H A P T E R 3 Steady-State Conduction—Multiple Dimensions 85
Table 3-1 (Continued).
Physical system Schematic Shape factor Restrictions
Isothermal cylinderof radius r placed insemi-infinite mediumas shown
2r
L
Isothermal2πL
ln(2L/r)L� 2r
Isothermal rectangularparallelepiped buried insemi-infinite medium havingisothermal surface
b
c
aL
Isothermal1.685L
[log
(1 + ba
)]−0.59 (bc
)−0.078 See Reference 7
Plane wall
L
A
A
L
One-dimensional heat flow
Hollow cylinder, length L
+
riro
2πL
ln(ro/ri)L� r
Hollow sphere
+
riro
4πroriro− ri
Thin horizontal disk buriedin semi-infinite mediumwith isothermal surface
Isothermal2r D
4r8r4πr
π/2 − tan−1(r/2D)
D= 0D� 2rD/2r> 1tan−1(r/2D) in radians
Hemisphere buried insemi-infinite medium�T = Tsphere − Tfar field
+r
Isothermal 2πr
Isothermal sphere buried insemi-infinite medium withinsulated surface +
r
D
T∞�
Insulated 4πr
1 + r/2D
Two isothermal spheresburied in infinite medium
+ +
r1 r2
D
4πr2r2r1
[1 − (r1/D)4
1−(r2/D)2]
− 2r2DD> 5rmax
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hol29362_Ch03 11/6/2008 9:46
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.86 K/PMS 293
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86 3-4 The Conduction Shape Factor
Table 3-1 (Continued).
Physical system Schematic Shape factor Restrictions
Thin rectangular plate oflengthL, buried in semi-infinitemedium having isothermalsurface
D
W
Isothermal
L
πW
ln(4W/L)
2πW
ln(4W/L)
2πW
ln(2πD/L)
D= 0W>L
D�WW>L
W �LD> 2W
Parallel disks buried ininfinite medium
D
t1 t2r
4πr[π2 − tan−1(r/D)
] D> 5rtan−1(r/D)in radians
Eccentric cylinders of length L
D
r1
r2
+ +
2πL
cosh−1(r21+r22−D2
2r1r2
) L� r2
Cylinder centered in a squareof length L
W
+Wr
L
2πL
ln(0.54W/r)L�W
Horizontal cylinder of length Lcentered in infinite plate
D
Isothermal
Isothermal
+r
D
2πL
ln(4D/r)
Thin horizontal disk buried insemi-infinite medium withadiabatic surface�T = Tdisk − Tfar field
D
Insulated
2r
4πr
π/2 + tan−1(r/2D)D/2r> 1tan−1(r/2D)in radians
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# 101675 Cust: McGraw-Hill Au: Holman Pg. No.92 K/PMS 293
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92 3-5 Numerical Method of Analysis
Table 3-2 Summary of nodal formulas for finite-difference calculations. (Dashed lines indicate element volume.)†
Nodal equation for equal increments in x and yPhysical situation (second equation in situation is in form for Gauss-Seidel iteration)
(a) Interior node
m, n + 1
m, n − 1
m, nm − 1, n m + 1, nΔy
Δy
Δx Δx
0 = Tm+1,n+ Tm,n+1 + Tm−1,n+ Tm,n−1 − 4Tm,n
Tm,n= (Tm+1,n+ Tm,n+1 + Tm−1,n+ Tm,n−1)/4
(b) Convection boundary node
m, n + 1
m, n − 1
m, nm − 1, n
Δy
Δy
Δx
h, T∞
0 = h�xkT∞ + 12 (2Tm−1,n+ Tm,n+1 + Tm,n−1)−
(h�xk
+ 2)Tm,n
Tm,n= Tm−1,n+(Tm,n+1+Tm,n−1)/2+Bi T∞2+BiBi = h�x
k
(c) Exterior corner with convection boundary
m, n − 1
m, nm − 1, n
Δy
Δx
h, T∞
0 = 2 h�xkT∞ + (Tm−1,n+ Tm,n−1)− 2
(h�xk
+ 1)Tm,n
Tm,n= (Tm−1,n + Tm,n−1)/2 + Bi T∞1+BiBi = h�x
k
(d) Interior corner with convection boundary
m, n − 1
m, n + 1
m, nm − 1, n m + 1, n
h, T∞Δy
Δx
0 = 2 h�xkT∞ + 2Tm−1,n+ Tm,n+1 + Tm+1,n+ Tm,n−1 − 2
(3 + h�x
k
)Tm,n
Tm,n= Bi T∞+Tm,n+1+Tm−1,n+(Tm+1,n + Tm,n−1)/23+BiBi = h�x
k
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C H A P T E R 3 Steady-State Conduction—Multiple Dimensions 93
Table 3-2 (Continued).
Nodal equation for equal increments in x and yPhysical situation (second equation in situation is in form for Gauss-Seidel iteration)
(e) Insulated boundary
m, n − 1
m, n + 1
m − 1, n
Δy
Δx
m, nIn
sula
ted
0 = Tm,n+1 + Tm,n−1 + 2Tm−1,n− 4Tm,nTm,n= (Tm,n+1 + Tm,n−1 + 2Tm−1,n)/4
(f ) Interior node near curved boundary‡
3
21
m, n + 1
m, n − 1
m, nm − 1, n m + 1, n
h,T∞�Δy
Δy
Δx
a Δx
Δx
c Δx
b Δx
0 = 2b(b+ 1) T2 + 2a+ 1Tm+1,n+ 2b+ 1Tm,n−1 + 2a(a+ 1) T1 − 2
(1a + 1b
)Tm,n
(g) Boundary node with convection alongcurved boundary—node 2 for (f ) above§
0 = b√a2 + b2 T1 +
b√c2 + 1T3 +
a+1bTm,n+ h�xk (
√c2 + 1 +
√a2 + b2)T∞
−[
b√a2 + b2 +
b√c2+1 +
a+1b
+ (√c2 + 1 +
√a2 + b2) h�x
k
]T2
†Convection boundary may be converted to insulated surface by setting h= 0 (Bi = 0).‡This equation is obtained by multiplying the resistance by 4/(a+ 1)(b+ 1)§This relation is obtained by dividing the resistance formulation by 2.
Nine-Node Problem EXAMPLE 3-5Consider the square of Figure Example 3-5. The left face is maintained at 100◦C and the top faceat 500◦C, while the other two faces are exposed to an environment at 100◦C:
h= 10 W/m2 · ◦C and k= 10 W/m · ◦CThe block is 1 m square. Compute the temperature of the various nodes as indicated inFigure Example 3-5 and the heat flows at the boundaries.
SolutionThe nodal equation for nodes 1, 2, 4, and 5 is
Tm+1,n+ Tm−1,n+ Tm,n+1 + Tm,n−1 − 4Tm,n= 0The equation for nodes 3, 6, 7, and 8 is given by Equation (3-25), and the equation for 9 is givenby Equation (3-26):
h�x
k= (10)(1)(3)(10)
= 13
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Figure 4-5 Temperature distribution in the semi-infinite solid with convection boundary condition.
1
0.1
0.01
0.001
0.00010 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6
0.05
0.1
0.2
0.3
0.5
1
23
∞
h( )1/2/kατ
(4 )1/2ατx
1−(T
−T
∞)/
(Ti −
T∞
) =
1− q
iq
Figure 4-6 Nomenclature for one-dimensional solids suddenly subjected to convectionenvironment at T∞: (a) infinite plate of thickness 2L; (b) infinite cylinder ofradius r0; (c) sphere of radius r0.
T0 = centerline temperature T0 = centerline axis temperature T0 = center temperature
(a) (b) (c)
LL
xr0
r0 r
r
++
149
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Fig
ure
4-7
Mid
plan
ete
mpe
ratu
refo
ran
infin
itepl
ate
ofth
ickn
ess
2L:(
a)fu
llsc
ale.
k�hL
1.0
0.8
0.7
0.6 0.5
0.4 0.30.2
0.1 0.06
0
1412
109
87
65
4
3
2.5
2.0 1.8
1.6 1.4
1.2
100
9080
7060
50
4540 35
30
25
20 18
16
01
23
46
810
1412
2216
1820
2426
2830
4050
6070
8090
100
110
130
150
200
300
400
500
600
700
1.0
0.7
0.5
0.3
0.4
0.2
0.1
0.07
0.05
0.04
0.03
0.02
0.00
7
0.01
0.00
50.
004
0.00
3
0.00
2
0.00
1
�L2
= F
oατ
0
i
θθ
=(T0−T∞)/(Ti−T∞)
(a)
150
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C H A P T E R 4 Unsteady-State Conduction 151
Figure 4-7 (Continued). (b) expanded scale for 0< Fo< 4, from Reference 2.
1.0
0.7
0.5
0.1
0.2
0.3
0.4
10 2 3 40.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2
0�
i =
(T0
− T
∞)�(
Ti −
T∞
) θ
θ
ατ = FoL2
100
1825
161087
1.4
1.6
1.8
2.0
2.5
3
4
5
6
k�hL
= 1
�Bi
(b)
If one considers the solid as behaving as a lumped capacity during the cooling or heatingprocess, that is, small internal resistance compared to surface resistance, the exponentialcooling curve of Figure 4-5 may be replotted in expanded form, as shown in Figure 4-13using the Biot-Fourier product as the abscissa. We note that the following parameters applyfor the bodies considered in the Heisler charts.
(A/V)inf plate = 1/L(A/V)inf cylinder = 2/r0(A/V)sphere = 3/r0
Obviously, there are many other practical heating and cooling problems of interest. Thesolutions for a large number of cases are presented in graphical form by Schneider [7], andreaders interested in such calculations will find this reference to be of great utility.
The Biot and Fourier Numbers
A quick inspection of Figures 4-5 to 4-16 indicates that the dimensionless temperatureprofiles and heat flows may all be expressed in terms of two dimensionless parameterscalled the Biot and Fourier numbers:
Biot number = Bi = hsk
Fourier number = Fo = ατs2
= kτρcs2
In these parameters s designates a characteristic dimension of the body; for the plate it isthe half-thickness, whereas for the cylinder and sphere it is the radius. The Biot numbercompares the relative magnitudes of surface-convection and internal-conduction resistances
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Fig
ure
4-8
Axi
ste
mpe
ratu
refo
ran
infin
itecy
linde
rof
radi
usr 0
:(a)
full
scal
e.
1.0
0.7
0.5
0.4
0.3
0.2
0.1
0.07
0.05
0.04
0.03
0.02
0.01
0.00
7
0.00
50.
004
0.00
3
0.00
2
0.00
135
030
020
015
014
013
012
011
010
090
8070
6050
4030
2628
2422
2018
1614
1210
86
43
21
0
100
90 8070
60
50
4035
30
2520 18 16
45
1412
10 98
76
5 4
k�hr0
3.5 3
.02.
5
2.01.8
1.6
1.41.2
1.0
0.8
0.6 0.5
0.4 0.30.2 0.1
0
0
i
θθ
=(T0−T∞)/(Ti−T∞)
(a)
ατ=
Fo
r 02
152
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C H A P T E R 4 Unsteady-State Conduction 153
Figure 4-8 (Continued). (b) expanded scale for 0< Fo< 4, from Reference 2.
1.0
0.7
0.5
0.1
0.2
0.3
0.4
10 2 3
4
3.0
2.5
3.5
40 0.2 0.4 0.6 0.8 1.0 1.2 1.6 1.8 2.0
0�
i = (T
0 −
T∞
)�(T
i − T
∞)
θθ
ατ = For0
2
100
20
2550
161412
89
7
6
k�hr
0 =
1�B
i
5
(b)
to heat transfer. The Fourier modulus compares a characteristic body dimension with anapproximate temperature-wave penetration depth for a given time τ.
A very low value of the Biot modulus means that internal-conduction resistance isnegligible in comparison with surface-convection resistance. This in turn implies that thetemperature will be nearly uniform throughout the solid, and its behavior may be approxi-mated by the lumped-capacity method of analysis. It is interesting to note that the exponentof Equation (4-5) may be expressed in terms of the Biot and Fourier numbers if one takesthe ratio V/A as the characteristic dimension s. Then,
hA
ρcVτ= hτ
ρcs= hsk
kτ
ρcs2= Bi Fo
Applicability of the Heisler Charts
The calculations for the Heisler charts were performed by truncating the infinite seriessolutions for the problems into a few terms. This restricts the applicability of the charts tovalues of the Fourier number greater than 0.2.
Fo = ατs2> 0.2
For smaller values of this parameter the reader should consult the solutions and charts givenin the references at the end of the chapter. Calculations using the truncated series solutionsdirectly are discussed in Appendix C.
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Fig
ure
4-9
Cen
ter
tem
pera
ture
for
asp
here
ofra
diusr 0
:(a)
full
scal
e.
00.
51.
0
1.0
0.7
0.5
0.4
0.3
0.2
0.1
0.07
0.05
0.04
0.03
0.02
0.01
0.00
7
0.00
50.
004
0.00
3
0.00
2
0.00
11.
52
2.5
35
67
910
1520
2530
3540
4550
9013
017
021
025
04
8 ατ�r
0=
Fo
100
9080
70
906050
4035
30 25
20 18
16
14 12 10-
3.5
2.8
2.6
2.4
2.2
2.0 1.8
1.6 1.4
1.2 1.0
0.75
0.05
0.5
0.4
0.30.2
0
9 8 76
54
45
k�hr
0
2
0
i
θθ
=(T0−T∞)/(Ti−T∞)
(a)
154
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Figure 4-9 (Continued). (b) expanded scale for 0< Fo< 3, from Reference 2.
1.0
0.7
0.5
0.1
0.2
0.3
0.4
0.5 1.0 1.5 2.0 2.5 3.000 0.1 0.20.05 0.35
0.5 0.75 1.0 1.2 1.6 2.0 2.4 2.8 3.0 3.5
0/
i = (T
0 −
T∞
)/(T
i − T
∞)
θθ
ατ = Fo0r2
100
25303550
18141210
89
7
6
5
4
k/hr
0 =
1/B
i
(b)
Figure 4-10 Temperature as a function of center temperature in aninfinite plate of thickness 2L, from Reference 2.
0.01 0.02 0.05 0.1 0.2 1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
20
0
3 5 10 20 50 1000.5
� 0
=(T
−T
∞)/
(T0
−T
∞)
θθ
=khL
1Bi
0.8
0.9
1.0
0.6
0.4
x/L = 0.2
155
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Figure 4-11 Temperature as a function of axis temperature in aninfinite cylinder of radius r0, from Reference 2.
0.01 0.02 0.05 0.1 0.2 1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
20
0
3 5 10 20 50 1000.5
/ 0
=(T
−T
∞)/
(T0
−T
∞)
θθ
=khr0
1Bi
0.8
0.9
1.0
0.6
0.4
r /r0 = 0.2
Figure 4-12 Temperature as a function of center temperature for asphere of radius r0, from Reference 2.
0.01 0.02 0.05 0.1 0.2 1.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
20
0
3 5 10 20 50 1000.5
=khr0
1Bi
r/r0 = 0.2
0.4
0.6
0.8
0.9
1.0
0θ θ=
(T−
T∞
)/(T
0−
T∞
)
156
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Figure 4-13 Temperature variation with time for solids that may betreated as lumped capacities: (a) 0
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Figure 4-13 (Continued).
0 0.02 0.04 0.06 0.08 0.10.9
0.91
0.92
0.93
0.94
0.95
0.96
0.97
0.98
0.99
1
(c)
BiFo = c�h(A/V)τ
θθi
Figure 4-14 Dimensionless heat loss Q/Q0 of an infinite plane of thickness 2L with time,from Reference 6.
10−5 10−4 10−3 10−2 10−1 1 10 102 103 104
= Fo Bi2h2
k2ατ
QQ0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
hL/k
=0.
001
0.00
20.
005
0.01
0.02
0.05 0.1
0.2
0.5
1 2 5
10 20 50
158
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C H A P T E R 4 Unsteady-State Conduction 159
Figure 4-15 Dimensionlesss heat loss Q/Q0 of an infinite cylinder of radius r0 with time,from Reference 6.
10−5 10−4 10−3 10−2 10−1 1 10 102 103 104
= Fo Bi2h2
k2ατ
QQ0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
hr0
/k=
0.00
10.
002
0.00
50.
010.
020.
05 0.1
0.2
0.5
1 2 5
10 20 50
Figure 4-16 Dimensionless heat loss Q/Q0 of a sphere of radius r0 with time, fromReference 6.
10−5 10−4 10−3 10−2 10−1 1 10 102 103 104
= Fo Bi2h2
k2ατ
QQ0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
hr0
/k=
0.00
10.
002
0.00
5
0.05 0.1
0.2
0.5
1 2 5
10 20 500.0
10.
02
Sudden Exposure of Semi-InfiniteSlab to Convection EXAMPLE 4-5
The slab of Example 4-4 is suddenly exposed to a convection-surface environment of 70◦C witha heat-transfer coefficient of 525 W/m2 · ◦C. Calculate the time required for the temperature toreach 120◦C at the depth of 4.0 cm for this circumstance.
SolutionWe may use either Equation (4-15) or Figure 4-5 for solution of this problem, but Figure 4-5 iseasier to apply because the time appears in two terms. Even when the figure is used, an iterativeprocedure is required because the time appears in both of the variables h
√ατ/k and x/(2
√ατ).
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C H A P T E R 4 Unsteady-State Conduction 173
displayed in Table 4-2, the most restrictive stability requirement (smallest�τ) is exhibitedby an exterior corner node, assuming all the convection nodes have the same value of Bi.
Table 4-2 Explicit nodal equations. (Dashed lines indicate element volume.)†
StabilityPhysical situation Nodal equation for �x = �y requirement
(a) Interior node
m, n − 1
m, n + 1
m − 1, nΔy
Δy
Δx Δx
m, n m + 1, n
Tp+1m,n = Fo
(Tpm−1,n+ T
pm,n+1 + T
pm+1,n+ T
pm,n−1
)+[1 − 4(Fo)]Tpm,n
Tp+1m,n = Fo
(Tpm−1,n+ T
pm,n+1 + T
pm+1,n+ T
pm,n−1
−4Tpm,n)
+ Tpm,n
Fo ≤ 14
(b) Convection boundary node
m, n + 1
m, n − 1
Δy
Δy
Δx
m, nm − 1, n h, T∞�
Tp+1m,n = Fo [2Tpm−1,n+ T
pm,n+1 + T
pm,n−1 + 2(Bi)T
p∞]+[1 − 4(Fo)− 2(Fo)(Bi)]Tpm,n
Tp+1m,n = Fo [2Bi (Tp∞ − Tpm,n)+ 2Tpm−1,n+ T
pm,n+1
+Tpm,n−1 − 4T
pm,n] + Tpm,n
Fo(2 + Bi)≤ 12
(c) Exterior corner with convectionboundary
m, n − 1
m − 1, n
Δy
Δx
m, n
h, T∞�
Tp+1m,n = 2(Fo) [Tpm−1,n+ T
pm,n−1 + 2(Bi)T
p∞]+[1 − 4(Fo)− 4(Fo)(Bi)]Tpm,n
Tp+1m,n = 2Fo [Tpm−1,n+ T
pm,n−1 − 2T
pm,n
+2Bi(Tp∞ − Tpm,n)] + Tpm,n
Fo(1 + Bi)≤ 14
(d) Interior corner with convectionboundary
m, n + 1
m, n − 1
Δy
Δx
m − 1, n m + 1, n
h, T∞�
m, n
Tp+1m,n = 23 (Fo) [2T
pm,n+1 + 2T
pm+1,n
+2Tpm−1,n+ T
pm,n−1 + 2(Bi)T
p∞]+[1 − 4(Fo)− 43 (Fo)(Bi)]T
pm,n
Tp+1m,n = (4/3)Fo [Tpm,n+1 + T
pm+1,n
+Tpm−1,n− 3T
pm,n+ Bi (Tp∞ − Tpm,n)] + Tpm,n
Fo(3 + Bi)≤ 34
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174 4-6 Transient Numerical Method
Table 4-2 (Continued).
StabilityPhysical situation Nodal equation for �x = �y requirement
(e) Insulated boundary
∆y
∆x
m − 1, n
m, n + 1
m, n − 1
m, n
Insu
late
d
Tp+1m,n = Fo [2Tpm−1,n+ T
pm,n+1 + T
pm,n−1]
+[1 − 4(Fo)]Tpm,nFo ≤ 14
†Convection surfaces may be made insulated by setting h= 0 (Bi = 0).
Table 4-3 Implicit nodal equations. (Dashed lines indicate volume element.)
Physical situation Nodal equation for �x = �y
(a) Interior node
∆x ∆x
∆y
∆ym – 1,n
m – 1,n
m,n m + 1,n
m + 1,n
[1 + 4(Fo)]Tp+1m,n − Fo(Tp+1m−1,n+ T
p+1m,n+1 + T
p+1m+1,n
+ Tp+1m,n−1
)− Tpm,n= 0
(b) Convection boundary node
∆x
∆y
∆ym – 1, n
m, n
m, n − 1
m, n + 1
h, T∞
[1 + 2(Fo)(2 + Bi)]Tp+1m,n − Fo[Tp+1m,n+1 + T
p+1m,n−1
+2Tp+1m−1,n+ 2(Bi)T
p+1∞]− Tpm,n= 0
(c) Exterior corner with convection boundary
∆x
∆y
m – 1, nm ,n
m, n − 1
h, T∞
[1 + 4(Fo)(1 + Bi)]Tp+1m,n − 2(Fo)[Tp+1m−1,n+ T
p+1m,n−1
+2(Bi)Tp+1∞]− Tpm,n= 0
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C H A P T E R 4 Unsteady-State Conduction 175
Table 4-3 (Continued).
Physical situation Nodal equation for �x = �y
(d) Interior corner with convectionboundary
Δx
Δy
m – 1, n m, n
m, n − 1
m, n + 1
m + 1, n
h, T∞�
[1 + 4(Fo)
(1 + Bi
3
)]Tp+1m,n − 2(Fo)3 ×
[2Tp+1m−1,n+ T
p+1m,n−1
+ 2Tp+1m,n+1 + 2T
p+1m+1,n+ 2(Bi)T
p+1∞]− Tpm,n= 0
(e) Insulated boundary
Δx
Δy
m – 1, n m, n
m, n − 1
m, n + 1
Insu
late
d
[1 + 4(Fo)]Tp+1m,n − Fo(
2Tp+1m−1,n+ T
p+1m,n+1 + T
p+1m,n−1
)− Tpm,n= 0
The advantage of an explicit forward-difference procedure is the direct calculationof future nodal temperatures; however, the stability of this calculation is governed by theselection of the values of �x and �τ. A selection of a small value of �x automaticallyforces the selection of some maximum value of �τ. On the other hand, no such restrictionis imposed on the solution of the equations that are obtained from the implicit formulation.This means that larger time increments can be selected to speed the calculation. The obviousdisadvantage of the implicit method is the larger number of calculations for each time step.For problems involving a large number of nodes, however, the implicit method may result inless total computer time expended for the final solution because very small time incrementsmay be imposed in the explicit method from stability requirements. Much larger incrementsin �τ can be employed with the implicit method to speed the solution.
Most problems will involve only a modest number of nodes and the explicit formulationwill be quite satisfactory for a solution, particularly when considered from the standpointof the more generalized formulation presented in the following section.
For a discussion of many applications of numerical analysis to transient heat conductionproblems, the reader is referred to References 4, 8, 13, 14, and 15.
It should be obvious to the reader by now that finite-difference techniques may beapplied to almost any situation with just a little patience and care. Very complicated problemsthen become quite easy to solve with only modest computer facilities. The use of MicrosoftExcel for solution of transient heat-transfer problems is discussed in Appendix D.
Finite-element methods for use in conduction heat-transfer problems are discussed inReferences 9 to 13. A number of software packages are available commercially.
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C H A P T E R 5 Principles of Convection 265
Table 5-2 Summary of equations for flow over flat plates. Properties evaluated at Tf = (Tw+ T∞)/2 unless otherwise noted.
Flow regime Restrictions Equation Equation number
Heat transfer
Laminar, local Tw= const, Rex < 5 × 105, Nux= 0.332 Pr1/3Re1/2x (5-44)0.6< Pr< 50
Laminar, local Tw= const, Rex < 5 × 105, Nux= 0.3387 Re1/2x Pr
1/3[1 +
(0.0468
Pr
)2/3]1/4 (5-51)Rex Pr> 100
Laminar, local qw= const, Rex < 5 × 105, Nux= 0.453 Re1/2x Pr1/3 (5-48)0.6< Pr< 50
Laminar, local qw= const, Rex < 5 × 105 Nux= 0.4637 Re1/2x Pr
1/3[1 +
(0.0207
Pr
)2/3]1/4 (5-51)
Laminar, average ReL < 5 × 105, Tw= const NuL= 2 Nux=L= 0.664 Re1/2L Pr1/3 (5-46)Laminar, local Tw= const, Rex < 5 × 105, Nux= 0.564(Rex Pr)1/2
Pr � 1 (liquid metals)Laminar, local Tw= const, starting at Nux= 0.332 Pr1/3 Re1/2x
[1 − ( x0x )3/4
]−1/3(5-43)
x= x0, Rex < 5 × 105,0.6< Pr< 50
Turbulent, local Tw= const, 5 × 105
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312 6-6 Summary
Table 6-8 Summary of forced-convection relations. (See text for property evaluation.)
Subscripts: b= bulk temperature, f = film temperature, ∞ = free stream temperature,w= wall temperature
Geometry Equation Restrictions Equation number
Tube flow Nud = 0.023 Re0.8d Prn Fully developed turbulent flow, (6-4a)n= 0.4 for heating,n= 0.3 for cooling,0.6< Pr< 100,2500
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C H A P T E R 6 Empirical and Practical Relations for Forced-Convection Heat Transfer 313
Table 6-8 (Continued).
Subscripts: b= bulk temperature, f = film temperature, ∞ = free stream temperature,w= wall temperature
Geometry Equation Restrictions Equation number
Flow across spheres Nudf = 0.37 Re0.6df Pr ∼ 0.7 (gases), 17
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A P P E N D I X
A TablesTable A-1 The error function.
x
2√
ατerf
x
2√
ατ
x
2√
ατerf
x
2√
ατ
x
2√
ατerf
x
2√
ατ
0.00 0.00000 0.76 0.71754 1.52 0.968410.02 0.02256 0.78 0.73001 1.54 0.970590.04 0.04511 0.80 0.74210 1.56 0.972630.06 0.06762 0.82 0.75381 1.58 0.974550.08 0.09008 0.84 0.76514 1.60 0.97636
0.10 0.11246 0.86 0.77610 1.62 0.978040.12 0.13476 0.88 0.78669 1.64 0.979620.14 0.15695 0.90 0.79691 1.66 0.981100.16 0.17901 0.92 0.80677 1.68 0.982490.18 0.20094 0.94 0.81627 1.70 0.98379
0.20 0.22270 0.96 0.82542 1.72 0.985000.22 0.24430 0.98 0.83423 1.74 0.986130.24 0.26570 1.00 0.84270 1.76 0.987190.26 0.28690 1.02 0.85084 1.78 0.988170.28 0.30788 1.04 0.85865 1.80 0.98909
0.30 0.32863 1.06 0.86614 1.82 0.989940.32 0.34913 1.08 0.87333 1.84 0.990740.34 0.36936 1.10 0.88020 1.86 0.991470.36 0.38933 1.12 0.88079 1.88 0.992160.38 0.40901 1.14 0.89308 1.90 0.99279
0.40 0.42839 1.16 0.89910 1.92 0.993380.42 0.44749 1.18 0.90484 1.94 0.993920.44 0.46622 1.20 0.91031 1.96 0.994430.46 0.48466 1.22 0.91553 1.98 0.994890.48 0.50275 1.24 0.92050 2.00 0.995322
0.50 0.52050 1.26 0.92524 2.10 0.9970200.52 0.53790 1.28 0.92973 2.20 0.9981370.54 0.55494 1.30 0.93401 2.30 0.9988570.56 0.57162 1.32 0.93806 2.40 0.9993110.58 0.58792 1.34 0.94191 2.50 0.999593
0.60 0.60386 1.36 0.94556 2.60 0.9997640.62 0.61941 1.38 0.94902 2.70 0.9998660.64 0.63459 1.40 0.95228 2.80 0.9999250.66 0.64938 1.42 0.95538 2.90 0.9999590.68 0.66278 1.44 0.95830 3.00 0.999978
0.70 0.67780 1.46 0.96105 3.20 0.9999940.72 0.69143 1.48 0.96365 3.40 0.9999980.74 0.70468 1.50 0.96610 3.60 1.000000
649
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Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.650 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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Tab
leA
-2Pr
oper
tyva
lues
for
met
als.
† Pro
pert
ies
at20
◦ CT
herm
alco
nduc
tivi
tyk,W
/m·◦
C
ρc p
kα
×10
5−1
00◦ C
0◦C
100◦
C20
0◦C
300◦
C40
0◦C
600◦
C80
0◦C
1000
◦ C12
00◦ C
Met
alkg
/m3
kJ/k
g·◦
CW
/m·◦
Cm
2 /s
−148
◦ F32
◦ F21
2◦F
392◦
F57
2◦F
752◦
F11
12◦ F
1472
◦ F18
32◦ F
2192
◦ F
Alu
min
um:
Pure
2,70
70.
896
204
8.41
821
520
220
621
522
824
9A
l-C
u(D
ural
umin
),94
–96%
Al,
3–5%
Cu,
trac
eM
g2,
787
0.88
316
46.
676
126
159
182
194
Al-
Si(S
ilum
in,
copp
er-b
eari
ng),
86.5
%A
l,1%
Cu
2,65
90.
867
137
5.93
311
913
714
415
216
1A
l-Si
(Alu
sil)
,78
–80%
Al,
20–2
2%Si
2,62
70.
854
161
7.17
214
415
716
817
517
8A
l-M
g-Si
,97%
Al,
1%M
g,1%
Si,
1%M
n2,
707
0.89
217
77.
311
175
189
204
Lea
d11
,373
0.13
035
2.34
336
.935
.133
.431
.529
.8Ir
on:
Pure
7,89
70.
452
732.
034
8773
6762
5548
4036
3536
Wro
ught
iron
,0.5
%C
7,84
90.
4659
1.62
659
5752
4845
3633
3333
Stee
l(C
max
≈1.
5%):
Car
bon
stee
lC
≈0.
5%7,
833
0.46
554
1.47
455
5248
4542
3531
2931
1.0%
7,80
10.
473
431.
172
4343
4240
3633
2928
291.
5%7,
753
0.48
636
0.97
036
3636
3533
3128
2829
650
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Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.651 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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Tab
leA
-2Pr
oper
tyva
lues
for
met
als†
(Con
tinu
ed).
Pro
pert
ies
at20
◦ CT
herm
alco
nduc
tivi
tyk
,W/m
·◦C
ρc p
kα
×10
5−1
00◦ C
0◦C
100◦
C20
0◦C
300◦
C40
0◦C
600◦
C80
0◦C
1000
◦ C12
00◦ C
Met
alkg
/m3
kJ/k
g·◦
CW
/m·◦
Cm
2 /s
−148
◦ F32
◦ F21
2◦F
392◦
F57
2◦F
752◦
F11
12◦ F
1472
◦ F18
32◦ F
2192
◦ F
Nic
kels
teel
Ni≈
0%7,
897
0.45
273
2.02
620
%7,
933
0.46
190.
526
40%
8,16
90.
4610
0.27
980
%8,
618
0.46
350.
872
Inva
r36
%N
i8,
137
0.46
10.7
0.28
6C
hrom
est
eel
Cr=
0%7,
897
0.45
273
2.02
687
7367
6255
4840
3635
361%
7,86
50.
4661
1.66
562
5552
4742
3633
335%
7,83
30.
4640
1.11
040
3836
3633
2929
2920
%7,
689
0.46
220.
635
2222
2222
2424
2629
Cr-
Ni(
chro
me-
nick
el):
15%
Cr,
10%
Ni
7,86
50.
4619
0.52
718
%C
r,8%
Ni
(V2A
)7,
817
0.46
16.3
0.44
416
.317
1719
1922
2731
20%
Cr,
15%
Ni
7,83
30.
4615
.10.
415
25%
Cr,
20%
Ni
7,86
50.
4612
.80.
361
Tun
gste
nst
eel
W=
0%7,
897
0.45
273
2.02
61%
7,91
30.
448
661.
858
5%8,
073
0.43
554
1.52
510
%8,
314
0.41
948
1.39
1C
oppe
r:Pu
re8,
954
0.38
3138
611
.234
407
386
379
374
369
363
353
Alu
min
umbr
onze
95%
Cu,
5%A
l8,
666
0.41
083
2.33
0
651
-
Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.652 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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Tab
leA
-2Pr
oper
tyva
lues
for
met
als†
(Con
tinu
ed).
Pro
pert
ies
at20
◦ CT
herm
alco
nduc
tivi
tyk,W
/m·◦
C
ρ,
c pk
α×
105
−100
◦ C0◦
C10
0◦C
200◦
C30
0◦C
400◦
C60
0◦C
800◦
C10
00◦ C
1200
◦ CM
etal
kg/m
3kJ
/kg
·◦C
W/m
·◦C
m2 /
s−1
48◦ F
32◦ F
212◦
F39
2◦F
572◦
F75
2◦F
1112
◦ F14
72◦ F
1832
◦ F21
92◦ F
Bro
nze
75%
Cu,
25%
Sn8,
666
0.34
326
0.85
9R
edbr
ass
85%
Cu,
9%Sn
,6%
Zn
8,71
40.
385
611.
804
5971
Bra
ss70
%C
u,30
%Z
n8,
522
0.38
511
13.
412
8812
814
414
714
7G
erm
ansi
lver
62%
Cu,
15%
Ni,
22%
Zn
8,61
80.
394
24.9
0.73
319
.231
4045
48C
onst
anta
n60
%C
u,40
%N
i8,
922
0.41
022
.70.
612
2122
.226
Mag
nesi
um:
Pure
1,74
61.
013
171
9.70
817
817
116
816
315
7M
g-A
l(el
ectr
o-ly
tic)
6–8%
Al,
1–2%
Zn
1,81
01.
0066
3.60
552
6274
83M
olyb
denu
m10
,220
0.25
112
34.
790
138
125
118
114
111
109
106
102
9992
Nic
kel:
Pure
(99.
9%)
8,90
60.
4459
902.
266
104
9383
7364
59N
i-C
r90
%N
i,10
%C
r8,
666
0.44
417
0.44
417
.118
.920
.922
.824
.680
%N
i,20
%C
r8,
314
0.44
412
.60.
343
12.3
13.8
15.6
17.1
18.0
22.5
Silv
er:
Pure
st10
,524
0.23
4041
917
.004
419
417
415
412
Pure
(99.
9%)
10,5
250.
2340
407
16.5
6341
941
041
537
436
236
0T
in,p
ure
7,30
40.
2265
643.
884
7465
.959
57T
ungs
ten
19,3
500.
1344
163
6.27
116
615
114
213
312
611
276
Zin
c,pu
re7,
144
0.38
4311
2.2
4.10
611
411
210
910
610
093
† Ada
pted
toSI
units
from
E.R
.G.E
cker
tand
R.M
.Dra
ke,H
eata
ndM
ass
Tran
sfer
,2nd
ed.N
ewY
ork:
McG
raw
-Hill
,195
9.
652
-
Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.653 K/PMS 293
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A P P E N D I X A Tables 653
Table A-3 Properties of nonmetals.†
Temperature k ρ c α × 107Substance ◦C W/m · ◦C kg/m3 kJ/kg · ◦C m2/s
Structural and heat-resistant materials
Acoustic tile 30 0.06 290 1.3 1.6Aluminum oxide,
sapphire 30 46 3970 0.76 150Aluminum oxide,
polycrystalline 30 36 3970 0.76 120Asphalt 20–55 0.74–0.76Bakelite 30 0.23 1200 1.6 1.2Brick:
Building brick,common 20 0.69 1600 0.84 5.2Face 1.32 2000
Carborundum brick 600 18.51400 11.1
Chrome brick 200 2.32 3000 0.84 9.2550 2.47 9.8900 1.99 7.9
Diatomaceous earth,molded and fired 200 0.24
870 0.31Fireclay brick 500 1.04 2000 0.96 5.4
Burnt 2426◦F 800 1.071100 1.09
Burnt 2642◦F 500 1.28 2300 0.96 5.8800 1.37
1100 1.40Missouri 200 1.00 2600 0.96 4.0
600 1.471400 1.77
Magnesite 200 3.81 1.13650 2.77
1200 1.90Cement, portland 0.29 1500
Mortar 23 1.16Coal,
anthracite 30 0.26 1300 1.25 1.6Concrete, cinder 23 0.76
Stone, 1-2-4 mix 20 1.37 1900–2300 0.88 8.2–6.8Glass, window 20 0.78 (avg) 2700 0.84 3.4
Corosilicate 30–75 1.09 2200Graphite, pyrolytic
parallel to layers 30 1900 2200 0.71 12,200perpendicular to
layers 30 5.6 2200 0.71 36Gypsum board 30 0.16Lexan 30 0.2 1200 1.3 1.3Nylon 30 0.16 1100 1.6 0.9Particle board,
low density 30 0.079 590 1.3 1.0high density 30 0.17 1000 1.3 1.3
Phenolic 30 0.03 1400 1.6 0.13Plaster, gypsum 20 0.48 1440 0.84 4.0
Metal lath 20 0.47Wood lath 20 0.28
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654 A P P E N D I X A Tables
Table A-3 Properties of nonmetals† (Continued).
Temperature k ρ c α × 107Substance ◦C W/m · ◦C kg/m3 kJ/kg · ◦C m2/s
Structural and heat-resistant materials
Plexiglass 30 0.2 1200 1.5 1.1Polyethylene 30 0.33 960 2.1 1.64Polypropylene 30 0.16 1150 1.9 0.73Polystrene 30 0.14 1000 1.3 1.1Polyvinylchloride 30 0.09 1700 1.1 0.48Rubber, hard 30 0.15 1200 2.0 0.62Silicon carbide 30 490 3150 0.68 2290Stone:
Granite 1.73–3.98 2640 0.82 8–18Limestone 100–300 1.26–1.33 2500 0.90 5.6–5.9Marble 2.07–2.94 2500–2700 0.80 10–13.6Sandstone 40 1.83 2160–2300 0.71 11.2–11.9
Structural concretelow density 30 0.21 670light weight 30 0.65 1570medium weight 30 0.75 1840normal weight 30 2.32 2260
Teflon 30 0.35 2200 1.05 1.5Titanium dioxide 30 8.4 4150 0.7 29Wood (across the grain):
Balsa, 8.8 lb/ft3 30 0.055 140Cypress 30 0.097 460Fir 23 0.11 420 2.72 0.96Maple or oak 30 0.166 540 2.4 1.28Yellow pine 23 0.147 640 2.8 0.82White pine 30 0.112 430
Insulating materials
Asbestos:Loosely packed −45 0.149
0 0.154 470–570 0.816 3.3–4100 0.161
Asbestos-cement 20 0.74boardsSheets 51 0.166Felt, 40 laminations/in 38 0.057
150 0.069260 0.083
20 laminations/in 38 0.078150 0.095260 0.112
Corrugated, 4 plies/in 38 0.08793 0.100
150 0.119Asbestos cement — 2.08
Balsam wood, 2.2 lb/ft3 32 0.04 35Cardboard, corrugated — 0.064
Celotex 32 0.048Cork, regranulated 32 0.045 45–120 1.88 2–5.3
Ground 32 0.043 150Corkboard, 10 lb/ft3 30 0.043 160
-
Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.655 K/PMS 293
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A P P E N D I X A Tables 655
Table A-3 Properties of nonmetals† (Continued).
Temperature k ρ c α × 107Substance ◦C W/m · ◦C kg/m3 kJ/kg · ◦C m2/s
Insulating materials
Diamond, Type IIa,insulator 30 2300 3500 0.509 12,900
Diatomaceous earth(Sil-o-cel) 0 0.061 320
Felt, hair 30 0.036 130–200Wool 30 0.052 330
Fiber, insulating board 20 0.048 240Glass fiber,
duct liner 30 0.038 32 0.84 14.1Glass fiber,
loose blown 30 0.043 16 0.84 32Glass wool, 1.5 lb/ft3 23 0.038 24 0.7 22.6Ice 0 2.22 910 1.93 12.6Insulex, dry 32 0.064
0.144Kapok 30 0.035Magnesia, 85% 38 0.067 270
93 0.071150 0.074204 0.080
Paper (avg.) 30 0.12 900 1.2 1.1Polyisocyanurate sheet 30 0.023Polystyrene, extruded 30 0.028Polyurethane foam 30 0.017Rock wool, 10 lb/ft3 32 0.040 160
Loosely packed 150 0.067 64260 0.087
Sawdust 23 0.059Silica aerogel 32 0.024 140Styrofoam 32 0.033Urethane, cerllular 30 0.025Wood shavings 23 0.059
†Adapted to SI units from A. I. Brown and S. M. Marco, Introduction to Heat Transfer, 3rd ed. New York: McGraw-Hill, 1958.Other properties from various sources.
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656 A P P E N D I X A Tables
Table A-4 Properties of saturated liquids.†
ρ cp k
T,◦C kg/m3 kJ/kg · ◦C ν, m2/s W/m · ◦C α, m2/s Pr β, K−1
Ammonia, NH3
−50 703.69 4.463 0.435×10−6 0.547 1.742×10−7 2.60−40 691.68 4.467 0.406 0.547 1.775 2.28−30 679.34 4.476 0.387 0.549 1.801 2.15−20 666.69 4.509 0.381 0.547 1.819 2.09−10 653.55 4.564 0.378 0.543 1.825 2.07
0 640.10 4.635 0.373 0.540 1.819 2.0510 626.16 4.714 0.368 0.531 1.801 2.0420 611.75 4.798 0.359 0.521 1.775 2.02 2.45 × 10−330 596.37 4.890 0.349 0.507 1.742 2.0140 580.99 4.999 0.340 0.493 1.701 2.0050 564.33 5.116 0.330 0.476 1.654 1.99
Carbon dioxide, CO2
−50 1,156.34 1.84 0.119×10−6 0.0855 0.4021×10−7 2.96−40 1,117.77 1.88 0.118 0.1011 0.4810 2.46−30 1,076.76 1.97 0.117 0.1116 0.5272 2.22−20 1,032.39 2.05 0.115 0.1151 0.5445 2.12−10 983.38 2.18 0.113 0.1099 0.5133 2.20
0 926.99 2.47 0.108 0.1045 0.4578 2.3810 860.03 3.14 0.101 0.0971 0.3608 2.8020 772.57 5.0 0.091 0.0872 0.2219 4.10 14.00 × 10−330 597.81 36.4 0.080 0.0703 0.0279 28.7
Sulfur dioxide, SO2
−50 1,560.84 1.3595 0.484×10−6 0.242 1.141×10−7 4.24−40 1,536.81 1.3607 0.424 0.235 1.130 3.74−30 1,520.64 1.3616 0.371 0.230 1.117 3.31−20 1,488.60 1.3624 0.324 0.225 1.107 2.93−10 1,463.61 1.3628 0.288 0.218 1.097 2.62
0 1,438.46 1.3636 0.257 0.211 1.081 2.3810 1,412.51 1.3645 0.232 0.204 1.066 2.1820 1,386.40 1.3653 0.210 0.199 1.050 2.00 1.94 × 10−330 1,359.33 1.3662 0.190 0.192 1.035 1.8340 1,329.22 1.3674 0.173 0.185 1.019 1.7050 1,299.10 1.3683 0.162 0.177 0.999 1.61
Dichlorodifluoromethane (Freon-12), CCl2F2
−50 1,546.75 0.8750 0.310×10−6 0.067 0.501×10−7 6.2 2.63 × 10−3−40 1,518.71 0.8847 0.279 0.069 0.514 5.4−30 1,489.56 0.8956 0.253 0.069 0.526 4.8−20 1,460.57 0.9073 0.235 0.071 0.539 4.4−10 1,429.49 0.9203 0.221 0.073 0.550 4.0
0 1,397.45 0.9345 0.214×10−6 0.073 0.557×10−7 3.810 1,364.30 0.9496 0.203 0.073 0.560 3.620 1,330.18 0.9659 0.198 0.073 0.560 3.530 1,295.10 0.9835 0.194 0.071 0.560 3.540 1,257.13 1.0019 0.191 0.069 0.555 3.550 1,215.96 1.0216 0.190 0.067 0.545 3.5
-
Hol29362_appA 11/7/2008 14:57
# 101675 Cust: McGraw-Hill Au: Holman Pg. No.657 K/PMS 293
Title: Heat Transfer 10/e Server: Short / Normal / Long
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A P P E N D I X A Tables 657
Table A-4 Properties of saturated liquids† (Continued).
ρ cp k
T,◦C kg/m3 kJ/kg · ◦C ν, m2/s W/m · ◦C α, m2/s Pr β, K−1
Glycerin, C3H5(OH)3
0 1,276.03 2.261 0.00831 0.282 0.983×10−7 84.7×10310 1,270.11 2.319 0.00300 0.284 0.965 31.020 1,264.02 2.386 0.00118 0.286 0.947 12.5 0.50 × 10−330 1,258.09 2.445 0.00050 0.286 0.929 5.3840 1,252.01 2.512 0.00022 0.286 0.914 2.4550 1,244.96 2.583 0.00015 0.287 0.893 1.63
Ethylene glycol, C2H4(OH)2
0 1,130.75 2.294 7.53×10−6 0.242 0.934×10−7 61520 1,116.65 2.382 19.18 0.249 0.939 204 0.65 × 10−340 1,101.43 2.474 8.69 0.256 0.939 9360 1,087.66 2.562 4.75 0.260 0.932 5180 1,077.56 2.650 2.98 0.261 0.921 32.4
100 1,058.50 2.742 2.03 0.263 0.908 22.4
Engine oil (unused)
0 899.12 1.796 0.00428 0.147 0.911×10−7 47,10020 888.23 1.880 0.00090 0.145 0.872 10,400 0.70 × 10−340 876.05 1.964 0.00024 0.144 0.834 2,87060 864.04 2.047 0.839×10−4 0.140 0.800 1,05080 852.02 2.131 0.375 0.138 0.769 490
100 840.01 2.219 0.203 0.137 0.738 276120 828.96 2.307 0.124 0.135 0.710 175140 816.94 2.395 0.080 0.133 0.686 116160 805.89 2.483 0.056 0.132 0.663 84
Mercury, Hg
0 13,628.22 0.1403 0.124×10−6 8.20 42.99×10−7 0.028820 13,579.04 0.1394 0.114 8.69 46.06 0.0249 1.82 × 10−450 13,505.84 0.1386 0.104 9.40 50.22 0.0207
100 13,384.58 0.1373 0.0928 10.51 57.16 0.0162150 13,264.28 0.1365 0.0853 11.49 63.54 0.0134
200 13,144.94 0.1570 0.0802 12.34 69.08 0.0116250 13,025.60 0.1357 0.0765 13.07 74.06 0.0103315.5 12,847 0.134 0.0673 14.02 81.5 0.0083
†Adapted to SI units from E. R. G. Eckert and R. M. Drake, Heat and Mass Transfer, 2nd ed. New York: McGraw-Hill, 1959.
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658 A P P E N D I X A Tables
Table A-5 Properties of air at atmospheric pressure.†
The values of µ, k, cp , and Pr are not strongly pressure-dependentand may be used over a fairly wide range of pressures
ρ cp µ × 105 ν × 106 k α × 104T,K kg/m3 kJ/kg · ◦C kg/m· s m2/s W/m · ◦C m2/s Pr
100 3.6010 1.0266 0.6924 1.923 0.009246 0.02501 0.770150 2.3675 1.0099 1.0283 4.343 0.013735 0.05745 0.753200 1.7684 1.0061 1.3289 7.490 0.01809 0.10165 0.739250 1.4128 1.0053 1.5990 11.31 0.02227 0.15675 0.722300 1.1774 1.0057 1.8462 15.69 0.02624 0.22160 0.708350 0.9980 1.0090 2.075 20.76 0.03003 0.2983 0.697400 0.8826 1.0140 2.286 25.90 0.03365 0.3760 0.689450 0.7833 1.0207 2.484 31.71 0.03707 0.4222 0.683500 0.7048 1.0295 2.671 37.90 0.04038 0.5564 0.680550 0.6423 1.0392 2.848 44.34 0.04360 0.6532 0.680600 0.5879 1.0551 3.018 51.34 0.04659 0.7512 0.680650 0.5430 1.0635 3.177 58.51 0.04953 0.8578 0.682700 0.5030 1.0752 3.332 66.25 0.05230 0.9672 0.684750 0.4709 1.0856 3.481 73.91 0.05509 1.0774 0.686800 0.4405 1.0978 3.625 82.29 0.05779 1.1951 0.689850 0.4149 1.1095 3.765 90.75 0.06028 1.3097 0.692900 0.3925 1.1212 3.899 99.3 0.06279 1.4271 0.696950 0.3716 1.1321 4.023 108.2 0.06525 1.5510 0.699
1000 0.3524 1.1417 4.152 117.8 0.06752 1.6779 0.7021100 0.3204 1.160 4.44 138.6 0.0732 1.969 0.7041200 0.2947 1.179 4.69 159.1 0.0782 2.251 0.7071300 0.2707 1.197 4.93 182.1 0.0837 2.583 0.7051400 0.2515 1.214 5.17 205.5 0.0891 2.920 0.7051500 0.2355 1.230 5.40 229.1 0.0946 3.262 0.7051600 0.2211 1.248 5.63 254.5 0.100 3.609 0.7051700 0.2082 1.267 5.85 280.5 0.105 3.977 0.7051800 0.1970 1.287 6.07 308.1 0.111 4.379 0.7041900 0.1858 1.309 6.29 338.5 0.117 4.811 0.7042000 0.1762 1.338 6.50 369.0 0.124 5.260 0.7022100 0.1682 1.372 6.72 399.6 0.131 5.715 0.7002200 0.1602 1.419 6.93 432.6 0.139 6.120 0.7072300 0.1538 1.482 7.14 464.0 0.149 6.540 0.7102400 0.1458 1.574 7.35 504.0 0.161 7.020 0.7182500 0.1394 1.688 7.57 543.5 0.175 7.441 0.730
†From Natl. Bur. Stand. (U.S.) Circ. 564, 1955.
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A P P E N D I X A Tables 659
Table A-6 Properties of gases at atmospheric pressure.†
Values of µ, k, cp , and Pr are not strongly pressure-dependent for He, H2, O2, and N2and may be used over a fairly wide range of pressures
ρ cp k
T, K kg/m3 kJ/kg · ◦C µ, kg/m · s v, m2/s W/m · ◦C α, m2/s Pr
Helium
144 0.3379 5.200 125.5×10−7 37.11×10−6 0.0928 0.5275×10−4 0.70200 0.2435 5.200 156.6 64.38 0.1177 0.9288 0.694255 0.1906 5.200 181.7 95.50 0.1357 1.3675 0.70366 0.13280 5.200 230.5 173.6 0.1691 2.449 0.71477 0.10204 5.200 275.0 269.3 0.197 3.716 0.72589 0.08282 5.200 311.3 375.8 0.225 5.215 0.72700 0.07032 5.200 347.5 494.2 0.251 6.661 0.72800 0.06023 5.200 381.7 634.1 0.275 8.774 0.72
Hydrogen
150 0.16371 12.602 5.595×10−6 34.18×10−6 0.0981 0.475×10−4 0.718200 0.12270 13.540 6.813 55.53 0.1282 0.772 0.719250 0.09819 14.059 7.919 80.64 0.1561 1.130 0.713300 0.08185 14.314 8.963 109.5 0.182 1.554 0.706350 0.07016 14.436 9.954 141.9 0.206 2.031 0.697400 0.06135 14.491 10.864 177.1 0.228 2.568 0.690450 0.05462 14.499 11.779 215.6 0.251 3.164 0.682500 0.04918 14.507 12.636 257.0 0.272 3.817 0.675550 0.04469 14.532 13.475 301.6 0.292 4.516 0.668600 0.04085 14.537 14.285 349.7 0.315 5.306 0.664700 0.03492 14.574 15.89 455.1 0.351 6.903 0.659800 0.03060 14.675 17.40 569 0.384 8.563 0.664900 0.02723 14.821 18.78 690 0.412 10.217 0.676
Oxygen
150 2.6190 0.9178 11.490×10−6 4.387×10−6 0.01367 0.05688×10−4 0.773200 1.9559 0.9131 14.850 7.593 0.01824 0.10214 0.745250 1.5618 0.9157 17.87 11.45 0.02259 0.15794 0.725300 1.3007 0.9203 20.63 15.86 0.02676 0.22353 0.709350 1.1133 0.9291 23.16 20.80 0.03070 0.2968 0.702400 0.9755 0.9420 25.54 26.18 0.03461 0.3768 0.695450 0.8682 0.9567 27.77 31.99 0.03828 0.4609 0.694500 0.7801 0.9722 29.91 38.34 0.04173 0.5502 0.697550 0.7096 0.9881 31.97 45.05 0.04517 0.641 0.700
Nitrogen
200 1.7108 1.0429 12.947×10−6 7.568×10−6 0.01824 0.10224×10−4 0.747300 1.1421 1.0408 17.84 15.63 0.02620 0.22044 0.713400 0.8538 1.0459 21.98 25.74 0.03335 0.3734 0.691500 0.6824 1.0555 25.70 37.66 0.03984 0.5530 0.684600 0.5687 1.0756 29.11 51.19 0.04580 0.7486 0.686700 0.4934 1.0969 32.13 65.13 0.05123 0.9466 0.691800 0.4277 1.1225 34.84 81.46 0.05609 1.1685 0.700900 0.3796 1.1464 37.49 91.06 0.06070 1.3946 0.711
1000 0.3412 1.1677 40.00 117.2 0.06475 1.6250 0.7241100 0.3108 1.1857 42.28 136.0 0.06850 1.8571 0.7361200 0.2851 1.2037 44.50 156.1 0.07184 2.0932 0.748
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660 A P P E N D I X A Tables
Table A-6 Properties of gases at atmospheric pressure† (Continued).
Values of µ, k, cp , and Pr are not strongly pressure-dependent for He, H2, O2, and N2and may be used over a fairly wide range of pressures
ρ cp k
T, K kg/m3 kJ/kg · ◦C µ, kg/m · s v, m2/s W/m · ◦C α, m2/s Pr
Carbon dioxide
220 2.4733 0.783 11.105×10−6 4.490×10−6 0.010805 0.05920×10−4 0.818250 2.1657 0.804 12.590 5.813 0.012884 0.07401 0.793300 1.7973 0.871 14.958 8.321 0.016572 0.10588 0.770350 1.5362 0.900 17.205 11.19 0.02047 0.14808 0.755400 1.3424 0.942 19.32 14.39 0.02461 0.19463 0.738450 1.1918 0.980 21.34 17.90 0.02897 0.24813 0.721500 1.0732 1.013 23.26 21.67 0.03352 0.3084 0.702550 0.9739 1.047 25.08 25.74 0.03821 0.3750 0.685600 0.8938 1.076 26.83 30.02 0.04311 0.4483 0.668
Ammonia, NH3
273 0.7929 2.177 9.353×10−6 1.18×10−5 0.0220 0.1308×10−4 0.90323 0.6487 2.177 11.035 1.70 0.0270 0.1920 0.88373 0.5590 2.236 12.886 2.30 0.0327 0.2619 0.87423 0.4934 2.315 14.672 2.97 0.0391 0.3432 0.87473 0.4405 2.395 16.49 3.74 0.0467 0.4421 0.84
Water vapor
380 0.5863 2.060 12.71×10−6 2.16×10−5 0.0246 0.2036×10−4 1.060400 0.5542 2.014 13.44 2.42 0.0261 0.2338 1.040450 0.4902 1.980 15.25 3.11 0.0299 0.307 1.010500 0.4405 1.985 17.04 3.86 0.0339 0.387 0.996550 0.4005 1.997 18.84 4.70 0.0379 0.475 0.991600 0.3652 2.026 20.67 5.66 0.0422 0.573 0.986650 0.3380 2.056 22.47 6.64 0.0464 0.666 0.995700 0.3140 2.085 24.26 7.72 0.0505 0.772 1.000750 0.2931 2.119 26.04 8.88 0.0549 0.883 1.005800 0.2739 2.152 27.86 10.20 0.0592 1.001 1.010850 0.2579 2.186 29.69 11.52 0.0637 1.130 1.019
†Adapted to SI units from E. R. G. Eckert and R. M. Drake, Heat and Mass Transfer, 2nd ed. New York: McGraw-Hill, 1959.
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A P P E N D I X A Tables 661
Table A-7 Physical properties of some common low-melting-point metals.†
NormalMelting boiling Density, Viscosity Heat Thermalpoint point Temperature ρ × 10−3 µ × 103 capacity conductivity Prandtl
Metal ◦C ◦C ◦C kg/m3 kg/m · s kJ/kg · ◦C W/m · ◦C number
Bismuth 271 1477 316 10.01 1.62 0.144 16.4 0.014760 9.47 0.79 0.165 15.6 0.0084
Lead 327 1737 371 10.5 2.40 0.159 16.1 0.024704 10.1 1.37 0.155 14.9 0.016
Lithium 179 1317 204 0.51 0.60 4.19 38.1 0.065982 0.44 0.42 4.19
Mercury −39 357 10 13.6 1.59 0.138 8.1 0.027316 12.8 0.86 0.134 14.0 0.0084
Potassium 63.8 760 149 0.81 0.37 0.796 45.0 0.0066704 0.67 0.14 0.754 33.1 0.0031
Sodium 97.8 883 204 0.90 0.43 1.34 80.3 0.0072704 0.78 0.18 1.26 59.7 0.0038
Sodium-potassium:22% Na 19 826 93.3 0.848 0.49 0.946 24.4 0.019
760 0.69 0.146 0.88356% Na −11 784 93.3 0.89 0.58 1.13 25.6 0.026
760 0.74 0.16 1.04 28.9 0.058Lead-bismuth,
44.5% Pb 125 1670 288 10.3 1.76 0.147 10.7 0.024649 9.84 1.15
†Adapted to SI units from J. G. Knudsen and D. L. Katz, Fluid Dynamics and Heat Transfer, New York: McGraw-Hill, 1958.
Table A-8 Diffusion coefficients of gases and vapors in air at 25◦C and 1 atm.†
Substance D, cm2/s Sc = νD
Substance D, cm2/s Sc = νD
Ammonia 0.28 0.78 Formic Acid 0.159 0.97Carbon dioxide 0.164 0.94 Acetic acid 0.133 1.16Hydrogen 0.410 0.22 Aniline 0.073 2.14Oxygen 0.206 0.75 Benzene 0.088 1.76Water 0.256 0.60 Toluene 0.084 1.84Ethyl ether 0.093 1.66 Ethyl benzene 0.077 2.01Methanol 0.159 0.97 Propyl benzene 0.059 2.62Ethyl alcohol 0.119 1.30
†From J. H. Perry (ed.), Chemical Engineers’ Handbook, 4th ed. New York: McGraw-Hill, 1963.
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662 A P P E N D I X A Tables
Table A-9 Properties of water (saturated liquid).†
Note: GrxPr =(
gβρ2cp
µk
)x3 �T
cp ρ µ kgβρ2cp
µk◦F ◦C kJ/kg · ◦C kg/m3 kg/m · s W/m · ◦C Pr 1/m3 · ◦C
32 0 4.225 999.8 1.79×10−3 0.566 13.2540 4.44 4.208 999.8 1.55 0.575 11.35 1.91 × 10950 10 4.195 999.2 1.31 0.585 9.40 6.34 × 10960 15.56 4.186 998.6 1.12 0.595 7.88 1.08 × 101070 21.11 4.179 997.4 9.8×10−4 0.604 6.78 1.46 × 101080 26.67 4.179 995.8 8.6 0.614 5.85 1.91 × 101090 32.22 4.174 994.9 7.65 0.623 5.12 2.48 × 1010
100 37.78 4.174 993.0 6.82 0.630 4.53 3.3 × 1010110 43.33 4.174 990.6 6.16 0.637 4.04 4.19 × 1010120 48.89 4.174 988.8 5.62 0.644 3.64 4.89 × 1010130 54.44 4.179 985.7 5.13 0.649 3.30 5.66 × 1010140 60 4.179 983.3 4.71 0.654 3.01 6.48 × 1010150 65.55 4.183 980.3 4.3 0.659 2.73 7.62 × 1010160 71.11 4.186 977.3 4.01 0.665 2.53 8.84 × 1010170 76.67 4.191 973.7 3.72 0.668 2.33 9.85 × 1010180 82.22 4.195 970.2 3.47 0.673 2.16 1.09 × 1011190 87.78 4.199 966.7 3.27 0.675 2.03200 93.33 4.204 963.2 3.06 0.678 1.90220 104.4 4.216 955.1 2.67 0.684 1.66240 115.6 4.229 946.7 2.44 0.685 1.51260 126.7 4.250 937.2 2.19 0.685 1.36280 137.8 4.271 928.1 1.98 0.685 1.24300 148.9 4.296 918.0 1.86 0.684 1.17350 176.7 4.371 890.4 1.57 0.677 1.02400 204.4 4.467 859.4 1.36 0.665 1.00450 232.2 4.585 825.7 1.20 0.646 0.85500 260 4.731 785.2 1.07 0.616 0.83550 287.7 5.024 735.5 9.51×10−5600 315.6 5.703 678.7 8.68
†Adapted to SI units from A. I. Brown and S. M. Marco, Introduction to Heat Transfer, 3rd ed. New York: McGraw-Hill, 1958.
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A P P E N D I X A Tables 663
Table A-10 Normal total emissivity of various surfaces.†
Surface T , ◦F Emissivity �
Metals and their oxides
Aluminum:Highly polished plate, 98.3% pure 440–1070 0.039–0.057Commercial sheet 212 0.09Heavily oxidized 299–940 0.20–0.31Al-surfaced roofing 100 0.216
Brass:Highly polished:
73.2% Cu, 26.7% Zn 476–674 0.028–0.03162.4% Cu, 36.8% Zn, 0.4% Pb, 0.3% Al 494–710 0.033–0.03782.9% Cu, 17.0% Zn 530 0.030
Hard-rolled, polished, but direction of polishing visible 70 0.038Dull plate 120–660 0.22Chromium (see nickel alloys for Ni-Cr steels), polished 100–2000 0.08–0.36
Copper:Polished 242 0.023
212 0.052Plate, heated long time, covered with thick oxide layer 77 0.78
Gold, pure, highly polished 440–1160 0.018–0.035Iron and steel (not including stainless):
Steel, polished 212 0.066Iron, polished 800–1880 0.14–0.38Cast iron, newly turned 72 0.44
turned and heated 1620–1810 0.60–0.70Mild steel 450–1950 0.20–0.32
Iron and steel (oxidized surfaces):Iron plate, pickled, then rusted red 68 0.61Iron, dark-gray surface 212 0.31Rough ingot iron 1700–2040 0.87–0.95Sheet steel with strong, rough oxide layer 75 0.80
Lead:Unoxidized, 99.96% pure 240–440 0.057–0.075Gray oxidized 75 0.28Oxidized at 300◦F 390 0.63
Magnesium, magnesium oxide 530–1520 0.55–0.20Molybdenum:
Filament 1340–4700 0.096–0.202Massive, polished 212 0.071
Monel metal, oxidized at 1110◦F 390–1110 0.41–0.46Nickel:
Polished 212 0.072Nickel oxide 1200–2290 0.59–0.86
Nickel alloys:Copper nickel, polished 212 0.059Nichrome wire, bright 120–1830 0.65–0.79Nichrome wire, oxidized 120–930 0.95–0.98
Platinum, polished plate, pure 440–1160 0.054–0.104Silver:
Polished, pure 440–1160 0.020–0.032Polished 100–700 0.022–0.031
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664 A P P E N D I X A Tables
Table A-10 Normal total emissivity of various surfaces† (Continued).
Surface T, ◦F Emissivity �
Metals and their oxides
Stainless steels:Polished 212 0.074Type 301; B 450–1725 0.54–0.63
Tin, bright tinned iron 76 0.043 and0.064
Tungsten, filament 6000 0.39Zinc, galvanized sheet iron, fairly bright 82 0.23
Refractories, building materials, paints, and miscellaneous
Alumina (85–99.5%, Al2O3, 0–12% SiO2, 0–1% Ge2O3);effect of mean grain size, microns (µm):10 µm 0.30–0.1850 µm 0.39–0.28100 µm 0.50–0.40
Asbestos, board 74 0.96Brick:
Red, rough, but no gross irregularities 70 0.93Fireclay 1832 0.75
Carbon:T-carbon (Gebrüder Siemens) 0.9% ash, started with 260–1160 0.81–0.79
emissivity of 0.72 at 260◦F but on heating changed tovalues given
Filament 1900–2560 0.526Rough plate 212–608 0.77Lampblack, rough deposit 212–932 0.84–0.78
Concrete tiles 1832 0.63Enamel, white fused, on iron 66 0.90Glass:
Smooth 72 0.94Pyrex, lead, and soda 500–1000 0.95–0.85
Paints, lacquers, varnishes:Snow-white enamel varnish on rough iron plate 73 0.906Black shiny lacquer, sprayed on iron 76 0.875Black shiny shellac on tinned iron sheet 70 0.821Black matte shellac 170–295 0.91Black or white lacquer 100–200 0.80–0.95Flat black lacquer 100–200 0.96–0.98Aluminum paints and lacquers:
10% Al, 22% lacquer body, on rough or smooth 212 0.52surfaceOther Al paints, varying age and Al content 212 0.27–0.67
Porcelain, glazed 72 0.92Quartz, rough, fused 70 0.93Roofing paper 69 0.91Rubber, hard, glossy plate 74 0.94Water 32–212 9.95–0.963
†Courtesy of H. C. Hottel, from W. H. McAdams, Heat Transmissions, 3rd ed. New York: McGraw-Hill, 1954.
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Table A-11 Steel-pipe dimensions.
Nominal Metal Inside cross-pipe Wall sectional sectional
size, in OD, in Schedule no. Thickness, in ID, in area, in2 area, ft2
18 0.405 40 0.068 0.269 0.072 0.00040
80 0.095 0.215 0.093 0.0002514 0.540 40 0.088 0.364 0.125 0.00072
80 0.119 0.302 0.157 0.0005038 0.675 40 0.091 0.493 0.167 0.00133
80 0.126 0.423 0.217 0.0009812 0.840 40 0.109 0.622 0.250 0.00211
80 0.147 0.546 0.320 0.0016334 1.050 40 0.113 0.824 0.333 0.00371
80 0.154 0.742 0.433 0.003001 1.315 40 0.133 1.049 0.494 0.00600
80 0.179 0.957 0.639 0.004991 12 1.900 40 0.145 1.610 0.799 0.01414
80 0.200 1.500 1.068 0.01225160 0.281 1.338 1.429 0.00976
2 2.375 40 0.154 2.067 1.075 0.0233080 0.218 1.939 1.477 0.02050
3 3.500 40 0.216 3.068 2.228 0.0513080 0.300 2.900 3.016 0.04587
4 4.500 40 0.237 4.026 3.173 0.0884080 0.337 3.826 4.407 0.7986
5 5.563 40 0.258 5.047 4.304 0.139080 0.375 4.813 6.122 0.1263
120 0.500 4.563 7.953 0.1136160 0.625 4.313 9.696 0.1015
6 6.625 40 0.280 6.065 5.584 0.200680 0.432 5.761 8.405 0.1810
10 10.75 40 0.365 10.020 11.90 0.547580 0.500 9.750 16.10 0.5185
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666 A P P E N D I X A Tables
Table A-12 Conversion factors. (See also inside cover.)
Length: Energy:12 in = 1 ft 1 ft · lbf = 1.356 J2.54 cm = 1 in 1 kWh = 3413 Btu1 µm = 10−6 m = 10−4 cm 1 hp · h = 2545 Btu
Mass: 1 Btu = 252 cal1 kg = 2.205 lbm 1 Btu = 778 ft · lbf1 slug = 32.16 lbm Pressure:454 g = 1 lbm 1 atm = 14.696 lbf /in2 = 2116 lbf /ft2
Force: 1 atm = 1.01325 × 105 Pa1 dyn = 2.248 × 10−6 lbf 1 in Hg = 70.73 lbf /ft21 lbf = 4.448 N Viscosity:105 dyn = 1 N 1 centipoise = 2.42 lbm/h · ft
1 lbf · s/ft2 = 32.16 lbm/s · ftThermal conductivity:
1 cal/s · cm · ◦C = 242 Btu/h · ft · ◦F1 W/cm · ◦C = 57.79 Btu/h · ft · ◦F
Useful conversion to SI units
Length: Volume:1 in = 0.0254 m 1 in3 = 1.63871 × 10−5 m31 ft = 0.3048 m 1 ft3 = 0.0283168 m31 mi = 1.60934 km 1 gal = 231 in3 = 0.0037854 m3
Area: Mass:1 in2 = 645.16 mm2 1 lbm= 0.45359237 kg1 ft2 = 0.092903 m2 Density:1 mi2 = 2.58999 km2 1 lbm/in3 = 2.76799 × 104 kg/m3
Pressure: 1 lbm/ft3 = 16.0185 × 104 kg/m31 N/m2 = 1 Pa Force:1 atm = 1.01325 × 105 Pa 1 dyn = 10−5 N1 lbf /in2 = 6894.76 Pa 1 lbf = 4.44822 N
Energy:1 erg = 10−7 J1 Btu = 1055.04 J1 ft · lbf = 1.35582 J1 cal (15◦C) = 4.1855 J
Power:1 hp = 745.7 W1 Btu/h = 0.293 W
Heat flux:1 Btu/h · ft2 = 3.15372 W/m21 Btu/h · ft = 0.96128 W/m
Thermal conductivity:1 Btu/h · ft · ◦F = 1.7307 W/m · ◦C
Heat-transfer coefficient:1 Btu/h · ft2 · ◦F = 5.6782 W/m2 · ◦C