quick reference for heat transfer analysis jason valentine...

Quick Reference for Heat Transfer Analysiscompiled by Jason Valentine and Greg WalkerPlease contact [email protected] with corrections and suggestions

copyleft 2018: You may copy, distribute, and modify and redistribute as long as you attribute the originalauthor and distribute with an equivalent license.

Version: October 9, 2018

Nomenclature:

symbol name unitsq heat transfer Wq′′ heat flux W/m2

q′′′ = g generation W/m3

k thermal conductivity W/m KE energy rate WR thermal resistance K/WG conductance W/KR′′ thermal resistance m2 K/Wh convection coefficient W/m2KU overall heat transfer coefficient W/m2Kθ temperature difference K (or ◦C)C = mcp thermal capacitance W/Kω periodic frequency rad/s

Conversions:

1 R-value = 1h ft2 ◦F

btu= 0.176

m2 K

W1 kW = 0.284 tons of refrigeration

Physical Constants:

name parameterStefan-Boltzmann σ = 5.670× 10−8 W/m2K4

Boltzmann kB = 1.381× 10−23 J/Kblackbody C1 = 2πhc20 = 3.742× 108 Wµm4/m2

blackbody C2 = hc0/kB = 1.439× 104 µm KWien’s C3 = 2898µm Kspeed of light c0 = 2.998× 108 m/sPlanck’s constant h = 6.626× 10−34 J s

Assumed values:

name parametersolar constant q′′solar = 1370 W/m2

solar temperature Tsolar = 5780 Kgravity g = 9.81 m/s2

critical Reynolds for flat plate Recr = 5× 105

critical Reynolds for inside tube Recr = 2300critical Rayleigh for vertical plate Racr = 109

radius of the Sun RSun = 6.957× 105 kmaverage distance to the Sun LSun = 150× 106 km

1 Conduction

1.1 Equivalent resistance

Assumptions:

1. steady state

2. constant properties

3. one-dimensional transport

4. no generation

expression cartesian cylindrical spherical

heat equationd2T

dx2= 0

1

r

d

dr

(rdT

dr

)= 0

1

r2d

dr

(r2dT

dr

)= 0

temperaturedistribution

T1 −∆Tx

LT2 + ∆T

ln(r/r2)

ln(r1/r2)T1 −∆T

1− (r1/r)

1− (r1/r2)

heat flux k∆T

L

k∆T

r ln(r2/r1)

k∆T

r2(1/r1 − 1/r2)

heat transfer kA∆T

L

2πLk∆T

ln(r2/r1)

4πk∆T

1/r1 − 1/r2

resistanceL

kA

ln(r2/r1)

2πkL

1/r1 − 1/r24πk

criticalradius

–k

h

2k

hr1 < r2; ∆T = T1 − T2

1.2 Shape Factors

Description schematic restrictions shape factor

isothermal sphere in asemi-infinite medium

T1

T2

z

D

z > D/22πD

1−D/4z

horizontal isothermalcylinder buried in asemi-infinite medium

T1

T2

z

L

D L� D2πL

cosh−1(2z/D)

2

vertical cylinder in asemi-infinite medium

T1

T2

L

D

L� D2πL

ln(4L/D)

two parallel cylinders oflength L in infinitemedium

D1

w

T2

D2T

1

L� D1, D2L� w

2πL

cosh−1(

4w2−D21−D2

2

2D1D2

)

circular cylinder of lengthL midway between twoparallel plates of equallength and infinite width

T2

T2

z

z

T1

D

z � D/2L� z

2πL

ln(8z/πD)

cylinder of length Lcentered in a square solidof equal length

T2

T1

D

ww > DL� w

2πL

ln(1.08w/D)

eccentric circular cylinderof length L in a cylinderof equal length

T2

T1

d

D

z

D > dL� D

2πL

cosh−1(D2+d2−4z2

2Dd

)

conduction through theedge of adjoining walls

T2

T1

D

L

D > 5L 0.54D

3

conduction through thecorner of three walls witha temperature difference∆T1−2 across the walls L

L� length and width 0.15L

disk of diameter D andtemperature T1 on asemi-infinite medium oftemperature T2

T2

T1

D

none 2D

square channel of lengthL

L

W

w

T1

2T

W/w < 1.42πL

0.785 ln(W/w)

W/w > 1.4 2πL

0.930 ln(W/w)− 0.050L�W

1.3 Temperature distribution for fins of uniform cross section

tip boundary θ(x)/θb heat transfer

θ(L) = θL(θL/θb) sinhβx+ sinhβ(L− x)

sinhβLkAcβθb

coshβL− θL/θbsinhβL

dθ

dx

∣∣∣∣x=L

= 0coshβ(L− x)

coshβLkAcβθb tanhβL

θ(L→∞) = 0 e−βx kAcβθb

dθ

dx

∣∣∣∣x=L

= −hkθ(L)

coshβ(L− x) + (h/βk) sinhβ(L− x)

coshβL+ (h/βk) sinhβLkAcβθb

sinhβL+ (h/βk) coshβL

coshβL+ (h/βk) sinhβL

β =√hP/kAc

θ ≡ T − T∞θb = θ(0) = T (0)− T∞θL = θ(L) = T (L)− T∞

4

1.4 Fin charts

1.4.1 Plate fins (or straight fins)

rectangle

w

2t

ηf =tanh ξ

ξ

triangle

w

2t

ηf =1

ξ

I1(2ξ)

I0(2ξ)

parabola

w

2t

ηf =2

1 +√

1 + 4ξ2

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 3.5 4 4.5 5

η

ξ = w√h/kt

rectangular fintriangular finparabolic fin

5

1.4.2 Spines (or pin fins)

rectangle

w

2t ηf =tanh

√2ξ√

2ξ

triangle

w

2t

ηf =2√2ξ

I2(2√

2ξ)

I1(2√

2ξ)

parabola

w

2t

ηf =2

1 +√

1 + 89ξ

2

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 3.5 4 4.5 5

η

ξ = w√h/kt

rectanglar fintrianglar fin

parabolic fin

6

1.4.3 Annular fins (or circular 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 3 3.5 4 4.5 5

η

ξ = w√h/kt

re/r

b = 1

re/r

b = 2

re/r

b = 3

re/r

b = 5

2t

rb

re

w

1.5 Semi-infinite slab solutions

penetration depth (90%): δp = 2.3√αt

1.5.1 Constant surface temperature T (0, t) = Ts

T (x, t)− TsTi − Ts

= erf

(x√4αt

)q′′s (t) =

k(Ts − Ti)√παt

1.5.2 Constant surface heat flux q′′(0, t) = q′′o

T (x, t)− Ti =q′′ok

[√4αt

πexp

(− x2

4αt

)− x erfc

(x√4αt

)]

1.5.3 Surface convection −k dT/dx|x=0 = h[T∞ − T (0, t)]

T (x, t)− TiT∞ − Ti

= erfc

(x√4αt

)− exp

(hx

k+h2αt

k2

)erfc

(x√4αt

+h√αt

k

)

7

1.5.4 Surface pulse of energy E

T (x, t)− Ti =E

ρc√παt

exp

(− x2

4αt

)

1.5.5 Steady-periodic

T (x, t)− Ti∆T

= exp(−x√ω/2α

)sin(ωt− x

√ω/2α

)penetration depth (90%): δp = 4

√α/ω

NB: ω has units of rad/s, not Hz.

1.5.6 Two semi-infinite slabs in contact

Ts =(√ρck)ATA,i + (

√ρck)BTB,i

(√ρck)A + (

√ρck)B

8

2 Convection

2.1 Forced/external: plate and sphere

Geometry Correlation Conditions

flat plate Nux = 0.332 Re1/2x Pr1/3 laminar, local, Tf , Pr > 0.6

flat plate NuL = 0.664 Re1/2L Pr1/3 laminar, average, Tf , Pr > 0.6

flat plate Nux = 0.0296 Re4/5x Pr1/3 turbulent, local, Tf , Rex < 108, 0.6 < Pr < 60

flat plate NuL = (0.037 Re4/5L − 871) Pr1/3

mixed, average, Tf , Recr = 5× 105,ReL < 108, 0.6 < Pr < 60

sphere NuD = 2 + (0.4 Re1/2D + 0.06 Re

2/3D ) Pr1/3

average, T∞, 3.5 < Re < 7.6× 104,0.71 < Pr < 380

falling drop NuD = 2 + 0.6 Re1/2D Pr1/3 average, T∞

Rex = U∞xν Nux = hx

kf

2.2 Forced/external: cylinder in a cross-flow

NuD =hD

k= C RemD Pr1/3 ReD =

U∞D

ν

ReD C m0.4–4 0.989 0.3304–40 0.911 0.385

40–4000 0.683 0.4664000–40,000 0.193 0.618

40,000–400,000 0.027 0.805 10-1

100

101

102

103

104

105

100

101

102

103

104

105

Nu

––

D

ReD

air

water

oil

2.3 Forced/external: non-circular bars

NuD =hD

k= C RemD Pr1/3

Geometry ReD C msquare

V 6000–60,000 0.304 0.59

V 5000–60,000 0.158 0.66thin plate

Vfront

back10,000–50,000 0.667 0.5007000–80,000 0.191 0.667

D is the height of the object

102

104

105

Nu

––

D/P

r1/3

ReD

diamondsquare

plate frontplate back

9

2.4 Forced/internal: turbulent flow in circular tubes


circular tube NuD = 0.023 Re4/5D Prn

turbulent, fully developed, 0.6 < Pr < 160,n = 0.4 for Ts > Tm, n = 0.3 for Ts < Tm

ReD = 4mπDµ Dh = 4Ac

P Recr = 2300

2.5 Forced/internal: fully developed laminar flow in tubes

section uniform qs uniform Tscircular 4.36 3.66rectangular see chart belowinfinite plates (b→∞) 8.23 7.54infinite plates (one sided) 5.39 4.86

2

3

4

5

6

7

1 2 3 4 5 6 7 8

Nusselt n

um

ber

(Nu

D)

aspect ratio (b/a)

uniform qsuniform Ts

b

a

10

2.6 Natural/external


vertical plate NuL = 0.59Ra1/4L laminar (104 < RaL < 109)

vertical plate NuL = 0.10Ra1/3L turbulent (109 < RaL < 1013)

horizontal plate NuL = 0.54Ra1/4L

upper surface of hot plate orlower surface of cold plate;104 < RaL < 107; Pr > 0.7


upper surface of hot plate orlower surface of cold plate;107 < RaL < 1011


lower surface of hot plate orupper surface of cold plate;104 < RaL < 109, Pr > 0.7

horizontal cylinder NuD =

{0.60 +

0.387Ra1/6D

[1 + (0.559/Pr)9/16]8/27

}2

RaD < 1012

sphere NuD = 2 +0.589Ra

1/4D

[1 + (0.469/Pr)9/16]4/9RaD < 1011; Pr > 0.7

Rax = gβ(Ts−T∞)x3

να

For 0 < θ < 60◦, where θ is the angle measured from vertical, use the vertical plate expression with greplaced by g cos θ.

2.7 Natural/internal


horizontal cavity NuL = 0.069Ra1/3L Pr0.074

heated from below;3× 105 < RaL < 7× 109

horizontal cavity NuL = 1 heated from above

vertical cavity NuL = 0.22

(Pr

0.2 + PrRaL

)0.28(H

L

)−1/4 2 < H/L < 10; Pr < 105;103 < RaL < 1010

vertical cavity NuL = 0.18

(Pr

0.2 + PrRaL

)0.29 1 < H/L < 2; 10−3 < Pr < 105;103 < RaLPr

0.2+Pr

vertical cavity NuL = 0.42Ra1/4L Pr0.012(H/L)−0.3

10 < H/L < 40; 1 < Pr < 2× 104;104 < RaL < 107

H is cavity height; L is distance between heated sides.

11

3 Radiation

3.1 Blackbody Emissive Power (band emission)

nλT F0→nλT nλT F0→nλT(µm K) (µm K)

200 0.0000 6200 0.7541400 0.0000 6400 0.7692600 0.0000 6600 0.7832800 0.0000 6800 0.7961

1000 0.0003 7000 0.80811200 0.0021 7200 0.81921400 0.0078 7400 0.82951600 0.0197 7600 0.83911800 0.0393 7800 0.84802000 0.0667 8000 0.85632200 0.1009 8500 0.87462400 0.1403 9000 0.89002600 0.1831 9500 0.90302800 0.2279 10000 0.91422898 0.2501 10500 0.92373000 0.2732 11000 0.93183200 0.3181 11500 0.93893400 0.3617 12000 0.94513600 0.4036 13000 0.95513800 0.4434 14000 0.96294000 0.4809 15000 0.96894200 0.5160 16000 0.97384400 0.5488 17000 0.97774600 0.5793 18000 0.98084800 0.6075 20000 0.98565000 0.6337 25000 0.99225200 0.6579 30000 0.99535400 0.6803 40000 0.99795600 0.7010 50000 0.99895800 0.7201 75000 0.99976000 0.7378 100000 0.9999

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1000 10000 100000

F0

-nλ

T

nλT (µm K)

10-10

10-5

100

105

1010

0.01 0.1 1 10 100 1000 10000

Draper point

0.0936

em

issiv

e p

ow

er

(W/m

2 µ

m)

wavelength (µm)

3 K77 K

303 K798 K

5780 KWien’s law

12

3.2 View factors

3.2.1 2D configurations (infinite into page)

Configuration Relation

w

w➋

➊

hH = h/w

F12 = F21 =√

1 +H2 −H

➋

➊

w

α

w

F12 = F21 = 1− sin(α

2

)

w

h

➋

➊

H = h/w

F12 =1

2

(1 +H −

√1 +H2

)

r r

➊➋s

X = 1 +s

2r

F12 = F21 =1

π

(√X2 − 1 + sin−1

1

X−X

)r➋

a

b

➊

c F12 =r

b− a

(tan−1

b

c− tan−1

a

c

)

13

3.2.2 3D configurations

Configuration Relation

h

y

x

➊

➋

0.01

0.1

1

0.1 1 10

y/h=0.1

0.2

0.4

1.0

2.0

10.0

F1

2

x/h

h

r2

r1

➋

➊

0

0.2

0.4

0.6

0.8

1

0.1 1 10

r2/h=0.3

0.5

0.7

1.0

1.5

2.0

3.04.0

6.010.0

F1

2

h/r1

l

h

w

➋

➊

0

0.1

0.2

0.3

0.4

0.5

0.1 1 10

w/l=10

4.0

2.0

1.0

0.6

0.4

0.2

0.1

0.05

F1

2

h/l

14

4 Heat exchanger relations

NTU =UA

Cmin; ε =

q

qmax; Cr =

Cmin

Cmax

Single stream (Cr = 0): ε = 1− exp(−NTU)

parallel counterflow

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00 0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00

shell and tube 1-pass shell and tube 2-pass

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00 0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00

crossflow (unmixed) crossflow (mixed)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00 0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

mixedmixed

dashed Cmaxdashed Cmax

solid Cminsolid Cmin

ε

NTU

Cr = 0.00

Cr = 0.25

Cr = 0.50

Cr = 0.75

Cr = 1.00

15

5 Material Properties

5.1 Solids

substance k (W/m K) cp (J/kg K) ρ (kg/m3) α (m2/s)aluminum 237 903 2702 97.1× 10−6

brick 0.72 835 1920 0.45× 10−6

cork 0.039 1800 120 0.18× 10−6

copper 401 385 8933 117× 10−6

diamond 2300 509 3500 1290× 10−6

fiberglass 0.038 835 32.0 0.142× 10−6

germanium 59.9 322 5360 34.7× 10−6

glass 1.38 745 2220 0.834× 10−6

gold 317 129 19300 127× 10−6

silicon 148 712 2330 89.2× 10−6

stainless 13.4 468 8238 3.48× 10−6

steel 60.5 434 7850 17.8× 10−6

styrofoam 0.040 1210 16.0 2.07× 10−6

tungsten 174 132 19300 68.3× 10−6

5.2 Fluids

substance k (W/m K) cp (J/kg K) ρ (kg/m3) ν (m2/s) α (m2/s) Prair 0.0263 1010 1.16 15.9× 10−6 22.5× 10−6 0.707freon 0.0803 1430 1200 0.145× 10−6 0.0468× 10−6 3.40mercury 8.54 139 13500 0.113× 10−6 4.55× 10−6 0.0248oil 0.145 1910 884 550× 10−6 0.0859× 10−6 6400water 0.613 4180 1000 0.858× 10−6 0.146× 10−6 5.83

5.3 Phase change

heat of fusion for water: hsf = 333.7 kJ/kgheat of vaporization for water: hfg = 2257 kJ/kg

16

6 Non-dimensional quantities

name expression description

Biot Bi =hL

ks

ratio of internal thermal resistance to the externalthermal resistance

Fourier Fo =αt

L2dimensionless time

Grashof Gr =gβ(Ts − T∞)L3

ν2ratio of buoyancy forces to viscous forces whereβ = 1/T

Jakob Ja =cp(Ts − Tsat)

hfgratio of sensible heat to latent energy

Nusselt Nu =hL

kfratio of convection to conduction in a fluid

Prandtl Pr =ν

α

ratio of momentum b.l thickness to thermal b.l.thickness

Rayleigh Ra =gβ(Ts − T∞)L3

να

ratio of thermally derived buoyancy to thermaldissipation (Ra = GrPr, where β = 1/T )

Reynolds Re =UL

νratio of inertial and viscous forces

7 Math

7.1 Solutions to ODEs

ODE solution

dy

dx+ λy = α y = C1 exp(−λx) +

α

λ

d2y

dx2− λ2y = α y = C1 exp(−λx) + C2 exp(λx)− α

λ2

y = C3 cosh(λx) + C4 sinh(λx)− α

λ2

d2y

dx2+ λ2y = α y = C1 cos(λx) + C2 sin(λx) +

α

λ2

17

7.2 Special functions

7.2.1 Error functions

0

0.2

0.4

0.6

0.8

1

0 1 2

f(x)

x

erferfc

7.2.2 Hyperbolic functions

-4

-3

-2

-1

0

1

2

3

4

-2 -1 0 1 2

f(x)

x

sinhcoshtanh

18

quick reference for heat transfer analysis jason valentine...

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