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# 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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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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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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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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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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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).


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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86 3-4 The Conduction Shape Factor



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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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


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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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.

hol29362_Ch04 10/30/2008 17:34



DESIGN SERVICES OF


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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DESIGN SERVICES OF



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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DESIGN SERVICES OF


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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DESIGN SERVICES OF


Figure 4-13 Temperature variation with time for solids that may betreated as lumped capacities: (a) 0

hol29362_ch04 10/14/2008 19:33



DESIGN SERVICES OF


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

hol29362_ch04 10/14/2008 19:33



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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).

hol29362_Ch04 10/30/2008 17:34



DESIGN SERVICES OF



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


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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DESIGN SERVICES OF


174 4-6 Transient Numerical Method


StabilityPhysical situation Nodal equation for �x = �y requirement


∆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


∆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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DESIGN SERVICES OF




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


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.

hol29362_Ch05 10/31/2008 14:59



DESIGN SERVICES OF


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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DESIGN SERVICES OF


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

hol29362_ch06 10/31/2008 14:58



DESIGN SERVICES OF


C H A P T E R 6 Empirical and Practical Relations for Forced-Convection Heat Transfer 313


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

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF


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

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF


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

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF


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



DESIGN SERVICES OF


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



DESIGN SERVICES OF


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

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF


654 A P P E N D I X A Tables

Table A-3 Properties of nonmetals† (Continued).


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



DESIGN SERVICES OF



Table A-3 Properties of nonmetals† (Continued).


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.

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF



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



DESIGN SERVICES OF



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.

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF



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.

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF



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

Hol29362_appA 11/7/2008 14:57



DESIGN SERVICES OF



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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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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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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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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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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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

useful conversion factors - mhriau.ac.irmhriau.ac.ir/_douranportal/documents/heat transfer...

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