Correlaciones Masa y Calor

Heat and Mass Correlations

Alexander Rattner, Jonathan Bohren

November 13, 2008

Contents

1 Dimensionless Parameters 2

2 Boundary Layer Analogies - Require Geometric Similarity 2

3 External Flow 33.1 External Flow for a Flat Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33.2 Mixed Flow Over a plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.3 Unheated Starting Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.4 Plates with Constant Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.5 Cylinder in Cross Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.6 Flow over Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.7 Flow Through Banks of Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.7.1 Geometric Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.7.2 Flow Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.8 Impinging Jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.9 Packed Beds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Internal Flow 94.1 Circular Tube . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4.1.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94.1.2 Flow Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.2 Non-Circular Tubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2 Flow Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.3 Concentric Tube Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3.1 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.3.2 Flow Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.4 Heat Transfer Enhancement - Tube Coiling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134.5 Internal Convection Mass Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Natural Convection 145.1 Natural Convection, Vertical Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.2 Natural Convection, Inclined Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Natural Convection, Horizontal Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.4 Long Horizontal Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.5 Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155.6 Vertical Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.7 Inclined Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.8 Rectangular Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.9 Concentric Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175.10 Concentric Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1

JRB, ASR MEAM333 - Convection Correlations

1 Dimensionless Parameters

Table 1: Dimensionless Parameters

αk

ρcpThermal diffusivity

Cfτs

ρu2∞/2

Skin Friction Coefficient

Leα

DABLewis Number - heat transfer vs. mass transport

NuhL

kfNusselt Number - Dimensionless Heat Transfer

Pe Pe = RexPr Peclet Number

Prν

α=µCpk

Prandtl Number - momentum diffusivity vs. thermal diffusivity

Reρu∞x

µ=u∞x

νReynolds Number - Inertia vs. Viscosity

Scν

DABSchmidt Number momentum vs. mass transport

ShhmL

DABSherwood Number - Dimensionless Mass Transfer

Sth

ρV cp=

NuLReLPr

Stanton Number - Modified Nusselt Number

StmhmV

=ShLReLSc

Stanton mass Number - Modified Sherwood Number

2 Boundary Layer Analogies - Require Geometric Similarity

Table 2: Boundary Layer Analogies

Heat and Mass Analogy

Nu

Prn=

Sh

Scn

hL

kPrn=

hmL

DABScn

Applies always for same geometry, n is positive

Chilton Colburn Heat jH =Cf2

= StPr2/3 0.6 < Pr < 60

Chilton Colburn Mass jM =Cf2

= StmSc2/3

0.6 < Sc < 3000

2/17


3 External Flow

These typically use properties at the film temperature Tf =Ts + T∞

2

3.1 External Flow for a Flat Plate

These use properties at the film temperature Tf =Ts + T∞

2

Table 3: Flat Plate Isothermal Laminar Flow

Flat plate Boundary Layer Thickness δ =5.0√u∞/vx

Re < 5E5

Local Shear Stress τs = 0.332u∞√ρµu∞/x Re < 5E5

Local Skin Friction Coefficient Cf,x = 0.664Re−0.5x Re < 1

Local Heat Transfer Nux =hxx

k= 0.332Re0.5x Pr1/3

Re < 5E5Pr ≥ 0.6

Local Mass Transfer Shx =hm,xx

DAB= 0.332Re0.5x Sc1/3

Re < 5E5Sc ≥ 0.6

Average Skin Friction Coefficient Cf,x = 1.328Re−0.5x Re < 1

Average Heat Transfer Nux =hxx

k= 0.664Re0.5x Pr1/3

IsothermalRe < 5E5Pr ≥ 0.6

Average Mass Transfer Shx =hm,xx

DAB= 0.664Re0.5x Sc1/3

Re < 5E5Sc ≥ 0.6

Nux Nux = 0.565Pe0.5x

Liquid MetalsNux = 2NuxPr ≤ 0.05Pex ≥ 100

Nux Nux =0.3387Re0.5x Pr1/3[

1 + (0.0468/Pr)2/3]1/4 All Prandtl Numbers

Pex ≥ 100

Table 4: Turbulent Flow Over an Isothermal Plate Rex > 5 · 105

Skin Friction Coefficient Cf,x = 0.0592Re−0.2x 5E5 < Re < 108

Boundary Layer Thickness δ = 0.37xRe−0.2x 5E5 < Re < 108

Heat Transfer Nux = StRexPr = 0.0296Re0.8x Pr1/35E5 < Re < 108

0.6 < Pr < 60

Mass Transfer Shx = StRexSc = 0.0296Re0.8x Sc1/35E5 < Re < 108

0.6 < Pr < 3000

3/17


3.2 Mixed Flow Over a plate

If transition occurs at xc

L ≥ 0.95 The laminar plate model may be used for h. Once the critical transition pointhas been found, we define A = 0.037Re0.8x,c − 0.664Re0.5x,c These typically use properties at the film temperature

Tf =Ts + T∞

2

Table 5: Mixed Flow Over an Isothermal Plate

Average Heat Transfer NuL = (0.037Re0.8L −A)Pr1/30.6 < Pr < 60

5 · 105 < ReL < 108

Average Skin Friction Coefficient CfL= 0.074Re−0.2 − 2A

ReL5 · 105 < ReL < 108

Average Mass Transfer ShL = (0.037Re0.8L −A)Sc1/30.6 < Sc < 60

5 · 105 < ReL < 108

3.3 Unheated Starting Length

Here the plate has Ts = T∞ until x = ζ These typically use properties at the film temperature Tf =Ts + T∞

2

Table 6: Unheated Starting Length

Local Heat Transfer Nux =Nux|ζ=0

[1− (ζ/x)0.75]1/3laminar0 < ReL < 5 · 105

Local Heat Transfer Nux =Nux|ζ=0[

1− (ζ/x)9/10]1/9 turbulent

5 · 105 < ReL < 108

Average Heat Transfer NuL = NuL|ζ=0LL−ζ

[1− (ζ/L)

p+1p+2

]p/(p+1) p = 2 Laminar Flowp = 8 Turbulent Flow

3.4 Plates with Constant Heat Flux

For average heat transfer values, it is acceptable to use the isothermal results for T =∫0L(Ts − T∞)dx

Table 7: Constant Heat Flux

Local Heat Transfer Laminar Nux = 0.453Re0.5x Pr1/30 < ReL < 5 · 105

Pr > 0.6

Local Heat Transfer Turbulent Nux = 0.0308Re0.8x Pr1/3ReL > 5 · 105

0.6 < Pr < 60

3.5 Cylinder in Cross Flow

For the cylinder in cross flow, we use ReD = ρV Dµ = V D

ν These typically use properties at the film temperature

Tf =Ts + T∞

2

4/17


Table 8: Cylinder in Cross Flow

NuD = CRemDPr1/3

0.7 < Pr < 60C,m are found as functions

of ReD on P426

NuD = CRemDPrn

(Pr

Prs

)0.25

0.7 < Pr < 5001 < ReD < 106

All properties evaluated atT∞ except Prs

Uses table 7.4 P428

NuD = 0.3 +0.62Re0.5D Pr1/3[

1 + (0.4/Pr)2/3]1/4

[1 +

(Red

282, 000

)5/8]4/5

Pr > 0.2

3.6 Flow over Spheres

Table 9: Flow over Spheres

NuD = 2 + (0.4Re0.5D + 0.06Re2/3D )Pr0.4(µ

µs

)1/4

0.71 < Pr < 3803.5 < Pr < 6.6 · 104

1.0 < (µ/µs) < 3.2All properties except µs

are evaluated at T∞NuD = 2 + 0.6Re0.5D Pr1/3 For Freely Falling Drops

NuD = 2Infinite Stationary Medium

Red → 0

5/17


3.7 Flow Through Banks of Tubes

3.7.1 Geometric Properties

Table 10: Tube Bank Properties

ReD =ρVmaxD

µ

Vmax =ST

ST −DVi

Aligned OR

Staggered and SD >ST +D

2Vmax =

ST2(SD −D)

Vi Staggered and SD <ST +D

2

Figure 1: Tube bank geometries for aligned (a) and staggered (b) banks

6/17


3.7.2 Flow Correlations

Table 11: Flow through banks of tubes

NuD = 1.13C1RemD,maxPr

1/3

More than 10 rows of tubes2000 < ReD,max < 40, 000

Pr > 0.7Coefficients come from

table 7.5 on P438

NuD|(NL<10) = C2NuD|(NL≥10)

C2 comes from Table 7.6 on P4392000 < ReD,max < 40, 000

Pr > 0.7Coefficients come from

table 7.5 on P438

NuD = CRemD,maxPr0.36

(Pr

Prs

)0.25

C,m comes from Table 7.7 on P4401000 < ReD,max < 2 · 106

0.7 < Pr < 500More than 20 rows

NuD|(NL<20) = C2NuD|(NL≥20)

For the above correlationC2 comes from Table 7.8 on P440

2000 < ReD,max < 40, 000Pr > 0.7

Table 12: Flow through banks of tubes 2

Log Mean Temp. ∆Tlm =(Ts − Ti)− (Ts − To)

ln(Ts−Ti

Ts−To

)Dimensionless Temp Correlation

Ts − ToTs − Ti

= exp

(− πDNh

ρV NTST cP

)N - total number of tubes, NT - total number of tubes in transverse plane

Heating Per Unit Length q′ = NhπD∆Tlm

7/17


3.8 Impinging Jets

Heat and mass transfer is measured against the fluid properties at the nozzle exit q′′ = h(Ts − Te) The Reynoldsand Nusselt numbers are measured using the hydraulic diameter of the nozzle Dh = Ac,e

P The Reynolds numberuses the nozzle exit velocity. All correlations use the target cell region Ar which is affected by the nozzle. This isdepicted in Figure 7.17 on P449. H is the height from the plate to the nozzle exit

Table 13: Impinging Jets

SingleRound Nozzle

Nu = Pr0.42G(Ar,

HD

) [2Re0.5(1 + 0.005Re0.55)0.5

] 2000 < Re < 4 · 105

2 < H/D < 120.004 < Ar < 0.04

G factor G = 2A0.5r

1− 2.2A0.5r

1 + 0.2(H/d− 6)Ar0.5Always

Round NozzleArray

Nu = Pr0.420.5K(Ar,

HD

)G(Ar,

HD

)Re2/3

2000 < Re < 105

2 < H/D < 120.004 < Ar < 0.04

K factor K =[1 +

(H/D

0.6/Ar1/2

)6]−0.05

Always

SingleSlot Nozzle

Nu = Pr0.423.06

0.5/Ar +H/W + 2.78Rem

3000 < Re < 9 · 104

2 < H/D < 100.025 < Ar < 0.125

m factor m = 0.695−

[(1

4Ar

)+(H

2W

)1.33

+ 3.06

]−1

Always

Slot NozzleArray

Nu = Pr0.4223A

3/4r,o

(2Re

Ar/Ar,o +Ar,o/Ar

)2/3

SHWL ≥ 11500 < Re < 4 · 104

2 < H/D < 800.008 < Ar < 2.5Ar,o

Ar,o Ar,o =[60 + 4

(H

2W − 2)2]−0.5

Always

8/17


3.9 Packed Beds

For packed beds, the heat transfer depends on the total particle surface area Ap,t

q = hAp,t∆Tlm

The outlet temperature can be determined from the log mean relation

Ts − ToTs − Ti

= exp

(− hAp,tρViAc,bcp

)For Spheres :

εjH = εjm = 2.06Re−0.575D

where Pr or Sc ≈ 0.7 and 90 < ReD < 4000 For non spheres multiply the right hand side by a factor - uniformcylinders of L = D use 0.71, for uniform cubes use 0.71

ε is the porosity and is typically 0.3 to 0.5.

4 Internal Flow

4.1 Circular Tube

4.1.1 Properties

Table 14: Flow Conditions

Mean Velocityum =

m

ρAc

ReDReD ≡

ρumD

µ=µmD

ν turbulent onset @ ReD ≈ 2300

Hydrodynamic Entry Length

(xfd,hD

)lam≈ 0.05ReD

10 ≤(xfd,h

D

)turb≤ 60

Velocity Profileu(r)um

= 2

[1−

(r

r0

)2]

Moody Friction Factor

f ≡ −(dp/dx)Dρu2

m/s

f =64ReD

f = 0.316Re−1/4D

SmoothReD ≤ 2× 104

f = 0.184Re−1/4D

SmoothReD ≥ 2× 104

f = (0.790ln(ReD)− 1.64)−2 Smooth3000 ≤ ReD ≤ 5× 106

Power for Pressure Drop P = (∆p)∀ ∀ =m

ρ

9/17


Table 15: Constant Surface Heat Flux

Convective Heat Transfer qconv = q′′s (PL) q′′s = constant

Mean Temperature Tm(x) = Tm,i +q′′sP

mcpx q′′s = constant

Table 16: Constant Surface Temperature

Convective Heat Transfer qconv = hAs∆Tlm Ts = constant

Log Mean Temperature

∆Tlm ≡∆To −∆Ti

ln(∆To/∆Ti)

∆To∆Ti

=Ts − Tm(x)Ts − Tm,i

= exp(−Pxhmcp

) Ts = constant

Table 17: Constant External Environment Temperature

Heat Transfer q = UAs∆Tlm T∞ = constant

Log Mean Temperature∆To∆Ti

=T∞ − Tm(x)T∞ − Tm,i

= exp(−UAsmcp

)T∞ = constant


Table 18: Fully Developed Flow In Circular Tubes

NuD ≡hD

k= 4.36

lamniar

fully developed

q′′s = constant

NuD ≡hD

k= 3.66

lamniar

fully developed

Ts = constant

10/17


Table 19: Laminar Entry Region Flow In Circular Tubes

NuD ≡hD

k= 3.66 +

0.0668(D/L)ReDPr1 + 0.04[(D/L)ReDPr]2/3

lamniarTs = constant(thermal entry length)OR(combined with Pr ≥ 5)

NuD ≡hD

k= 1.86

(ReDPr

L/D

)1/3(µ

µs

)0.14

lamniarTs = constant0.60 ≤ Pr ≤ 5

0.0044 ≤(µ

µs

)≤ 9.75

All properties evaluated at the mean temperature Tm = (Tm,i + Tm,o)/2

Table 20: Turbulent Flow In Circular Tubes

NuD ≡hD

k= 0.023Re4/5D Prn

Ts > Tm : n = 0.4Ts < Tm : n = 0.3

turbulentfully developedsmall temperature diff0.6 ≤ Pr ≤ 160ReD ≥ 10, 000

NuD ≡hD

k= 0.027Re4/5D Pr1/3

(µ

µs

)0.14

laminar0.7 ≤ Pr ≤ 16, 700ReD ≥ 10, 000L

D≥ 10

NuD ≡hD

k=

(f/8)(ReD − 1000)Pr1 + 12.7(f/8)1/2(Pr2/3 − 1)

lamniar0.5 ≤ Pr ≤ 20003000 ≤ ReD ≤ 5× 106

Above appropriate for both constant Ts and constant q′′s

NuD ≡hD

k= 4.82 + 0.0185Pe0.827D

lamniarNOT liquid metals (3× 10−3 ≤ Pr ≤ 5× 10−2)q′′s = constant

3.6× 103 ≤ ReD ≤ 9.05× 105

102 ≤ PeD ≤ 104

NuD ≡hD

k= 5.0 + 0.025Pe0.8D

similarly as immediately aboveTs = constant

100 ≤ PeDAll properties evaluated at the mean temperature Tm = (Tm,i + Tm,o)/2

11/17


4.2 Non-Circular Tubes

4.2.1 Properties

Table 21: Flow in Non-Circular Tubes

Hydrodynamic Diameter Dh ≡4AcP

ReDh

ReDh≡ ρumDh

µ=µmDh

ν turbulent onset @ ReDh≈ 2300

All properties evaluated at the mean temperature Tm = (Tm,i + Tm,o)/2


Figure 2: Nusselt numbers and friction factors for fully developed laminar flow in tubes of differing cross-section

12/17


4.3 Concentric Tube Annulus

4.3.1 Properties

Table 22: Concentric Tube Annulus Properties

Interior heat transfer q′′i = hi(Ts,i − Tm)Exterior heat transfer q′′o = ho(Ts,o − Tm)Hydrodynamic Diameter Dh = Do −Di


Table 23: Correlations for Concentric Tube Annulus

See Table 8.2 on Page 520

lamniarfully developedone surface insulatedone surface const Ts

Nui =Nuii

1− (q′′o /q′′i )θ∗i, Nuo =

Nuoo1− (q′′i /q′′o )θ∗o

See Table 8.3 for above parameters as a function of Di

Do

laminarq′′i = constantq′′o = constant

4.4 Heat Transfer Enhancement - Tube Coiling

Table 24: Properties for Helically Coiled Tubes

CriticalReynolds Number

ReD,c,h = ReD,c[1 + 12(D/C)0.5]ReD,c = 2300

D,C are definedin Figure 8.13on Page 522

f f =64ReD

ReD(D/C)1/2 ≤ 30

f f =27

Re0D.725(D/C)0.1375 30 ≤ ReD(D/C)1/2 ≤ 300

f f =7.2

Re0D.5(D/C)0.25 300 ≤ ReD(D/C)1/2

Table 25: Correlations for Helically Coiled Tubes

NuD =

[(3.66 +

4.343a

)3

+ 1.158(ReD(D/C)1/2

b

)3/2]1/3(

µ

µs

)0.14

a =(

1 +927(C/D)Re2DPr

)

b = 1 +0.477Pr

0.005 ≤ Pr ≤ 1600

1 ≤ ReDDC

1/2 ≤ 1000

13/17


4.5 Internal Convection Mass Transfer

Table 26: Properties for Internal Convection Mass Transfer

MeanSpecies Density ρA,m =

∫Ac

(ρAu)dAcumAc

Any Shape

MeanSpecies Density

ρA,m =2

umr2o

∫ ro

0(ρAur)dr Circular Tube

LocalMass Flux

n′′A = hm(ρA,s − ρA,m)

TotalMass Flux

nA = hmAs∆ρA,lm

nA =m

ρ(ρA,o − ρA, i)

Log MeanConcentration Difference

∆ρA,lm =∆ρA,o −∆ρA,i

ln(∆ρA,o/∆ρA,i)

∆ρA(x)∆ρA,i

=ρA,s − ρA,m(x)ρA,s − ρA,m,i

= exp(−hmρP

mx

)

Sherwood Number

ShD =hmD

DAB

ShD =hmD

DAB

The concentration entry length xfd,c can be determined with the mass transfer analogy and the same functionused to determine xfd,t. From this point, the appropriate heat transfer correlation can be invoked along the linesof the mass transfer analogy,

5 Natural Convection

Natural Convection uses the Rayleigh number instead of the Reynolds number. Transition to turbulent flowhappens around

Ra ≈ 109

14/17


5.1 Natural Convection, Vertical Plate

Table 27: Natural Convection, Vertical Plate

Laminar Heat Transfer Nux =(Grx

4

)1/4

g(Pr) uses g below

g factor g(Pr) =0.75Pr0.5

(0.609 + 1.221Pr0.5 + 1.238Pr)1/40 < Pr <∞

Average Laminar NuL =43

(Grx

4

)1/4

g(Pr) laminar

Better avg. Heat Transfer NuL =

[0.825 +

0.387Ra1/6l[

1 + (0.492/Pr)9/16]8/27

]2

Applies for all RaL

Better avg. Laminar Heat Transfer NuL = 0.68 +0.670Ra1/4

l[1 + (0.492/Pr)9/16

]4/9 RaL < 109

5.2 Natural Convection, Inclined Plate

For the top of a cooled plate and the bottom of a heated plates, the vertical correlations can be used with g cos(θ)substituted into RaL for a tilt of up to 60 degrees away from the vertical (0 = vertical). No recommendations arerecommended for the other cases.

5.3 Natural Convection, Horizontal Plate

These correlations use L = As

P

Table 28: Natural Convection, Horizontal Plate

Upper Surface Hot PlateLower Surface Cold Plate

NuL = 0.54Ra1/4L 104 < RaL < 107

Upper Surface Hot PlateLower Surface Cold Plate

NuL = 0.15Ra1/3L 107 < RaL < 1011

Lower Surface Hot PlateUpper Surface Cold Plate

NuL = 0.27Ra1/4L 105 < RaL < 1010

5.4 Long Horizontal Cylinder

Assumes isothermal cylinder. The following correlation applies for RaD < 1012

NuD =

[0.60 +

0.387Ra1/6D[

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

]2

5.5 Spheres

For Pr > 0.7 and RaD < 1011

NuD = 2 +0.589Ra1/4

D[1 + (0.469/Pr)9/16

]4/9

15/17


5.6 Vertical Channels

This section describes correlations for natural convection between to parralel plates. It uses Ras which uses theplate separation for the length scale. I believe that the convection area is the surface area where heating/coolinghappens.

Table 29: Vertical Channels

Symmetrically HeatedIsothermal Plates

Nus = 124Ras

(S

L

)[1− exp

(− 35Ras(S/L)

)]0.7510−1 < S

LRas < 105

Symmetrically HeatedIsothermal Plates

Nus =RAs(S/L)

2410−1 < S

LRas < 105

SL → 0

1 Insulated Plate2 Isothermal Plate

Nus =Ras(S/L)

1210−1 < S

LRas < 105

SL → 0

Isothermal /Adiabatic(Better)

Nus =[

C1

(RasS/L)2+

C2

(RasS/L)1/2

]−1/2

RasSL ≤ 10

The isothermal correlations use Nus =(

q/A

Ts − T∞

)S

kand Ras =

gβ(Ts − T∞)S3

αν

The better isothermal correlation usesC1 = 576, C2 = 2.87 for Symmetric isothermal PlatesC1 = 144, C2 = 2.87 for isothermal and adiabatic Plates

SymmetricIsoflux Plates

Nus,L,fd = 0.144 [Ra∗s(S/L)]0.5 Uses Ra∗

1 Isoflux Plate1 Insulated

Nus,L,fd = 0.204 [Ra∗s(S/L)]0.5 Uses Ra∗

Isoflux /Adiabatic(Better)

Nus,L =[

C1

Ra∗sS/L+

C2

(Ra∗sS/L)2/5

]−1/2

RasSL ≥ 100

The isoflux corelations use Nus,fd =(

q′′sTs,L − T∞

)S

kand Ra∗s =

gβq′′sS4

kαν

The better isoflux correlation usesC1 = 48, C2 = 2.51 for Symmetric isoflux PlatesC1 = 24, C2 = 2.51 for isoflux and adiabatic Plates

5.7 Inclined Channels

For plates inclined less than 45 degrees from the vertical

Nus = 0.645 [Ras(S/L)]1/4

Fluid properties are evaluated at T = Ts+T∞2 This requires Ras(S/L) > 200

5.8 Rectangular Cavities

For a channel with flow through the HxL plane, no advection happens unless

RaL > 1708

See Figure 9.10 on p 588 for geometric details All properties are evaluated at the average between the heattransferring plates. Inclined plates are discussed on P590.

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Table 30: Rectangular Channels

Horizontal CavityHeated from Below

NuL = 0.069Ra1/3L Pr0.074

3 · 105 < RaL < 7 · 109

All properties evaluated ataverage temp. betweenhot and cold plates

Heat transfer onVertical Surfaces

NuL = 0.22(

Pr

0.2 + PrRaL

)0.28(H

L

)−0.25 103 < RaL < 10102 ≤ H

L ≤ 10Pr ≤ 105


NuL = 0.18(

Pr

0.2 + PrRaL

)0.29 103 < RaLPr0.2+Pr

1 ≤ HL ≤ 2

10−3 ≤ Pr ≤ 105


NuL = 0.42Ra0.25L Pr0.012

(H

L

)−0.3 104 < RaL < 107

10 ≤ HL ≤ 40

1 ≤ Pr ≤ 2 · 104


NuL = 0.046Ra1/3L

106 < RaL ≤ 109

1 ≤ HL ≤ 40

1 ≤ Pr ≤ 20

5.9 Concentric Cylinders

For Cylinders we use an effective thermal conductivity

keffk

= 0.386(

Pr

0.861 + Pr

)1/4

Ra1/4c

The Rayleigh number uses the corrected length

Lc =2 [ln(ro/ri)]

4/3

(r−0.6i + r−0.6

o )5/3

The Heat Transfer is found as

q =2πLkeff (Ti − To)

ln(ro/ri)

5.10 Concentric Spheres

For Spheres we use an effective thermal conductivity

keffk

= 0.74(

Pr

0.861 + Pr

)1/4

Ra1/4s

The Rayleigh number uses the corrected length

Ls =

(1ri− 1

ro

)4/3

21/3(r−7/5i + r

−7/5o )5/3

The Heat Transfer is found as

q =4πLkeff (Ti − To)

(1/ri)− (1/ro)

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