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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
4.1.2 Flow Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
4.2.2 Flow Correlations
Figure 2: Nusselt numbers and friction factors for fully developed laminar flow in tubes of differing cross-section
12/17
JRB, ASR MEAM333 - Convection Correlations
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
4.3.2 Flow Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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
JRB, ASR MEAM333 - Convection Correlations
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.
16/17
JRB, ASR MEAM333 - Convection Correlations
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
Heat transfer onVertical Surfaces
NuL = 0.18(
Pr
0.2 + PrRaL
)0.29 103 < RaLPr0.2+Pr
1 ≤ HL ≤ 2
10−3 ≤ Pr ≤ 105
Heat transfer onVertical Surfaces
NuL = 0.42Ra0.25L Pr0.012
(H
L
)−0.3 104 < RaL < 107
10 ≤ HL ≤ 40
1 ≤ Pr ≤ 2 · 104
Heat transfer onVertical Surfaces
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)
17/17