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
3.1 External Flow for a Flat Plate .
3.2 Mixed Flow Over a plate . . . . .
3.3 Unheated Starting Length . . . .
3.4 Plates with Constant Heat Flux .
3.5 Cylinder in Cross Flow . . . . . .
3.6 Flow over Spheres . . . . . . . .
3.7 Flow Through Banks of Tubes .
3.7.1 Geometric Properties . .
3.7.2 Flow Correlations . . . .
3.8 Impinging Jets . . . . . . . . . .
3.9 Packed Beds . . . . . . . . . . . .
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4 Internal Flow
4.1 Circular Tube . . . . . . . . . . . .
4.1.1 Properties . . . . . . . . . .
4.1.2 Flow Correlations . . . . .
4.2 Non-Circular Tubes . . . . . . . .
4.2.1 Properties . . . . . . . . . .
4.2.2 Flow Correlations . . . . .
4.3 Concentric Tube Annulus . . . . .
4.3.1 Properties . . . . . . . . . .
4.3.2 Flow Correlations . . . . .
4.4 Heat Transfer Enhancement - Tube
4.5 Internal Convection Mass Transfer
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3
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Coiling
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9
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14
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14
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16
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17
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5 Natural Convection
5.1 Natural Convection, Vertical Plate .
5.2 Natural Convection, Inclined Plate .
5.3 Natural Convection, Horizontal Plate
5.4 Long Horizontal Cylinder . . . . . .
5.5 Spheres . . . . . . . . . . . . . . . .
5.6 Vertical Channels . . . . . . . . . . .
5.7 Inclined Channels . . . . . . . . . . .
5.8 Rectangular Cavities . . . . . . . . .
5.9 Concentric Cylinders . . . . . . . . .
5.10 Concentric Spheres . . . . . . . . . .
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1
JRB, ASR
1
MEAM333 - Convection Correlations
Dimensionless Parameters
Table 1: Dimensionless Parameters
α
Cf
Le
Nu
Pe
Pr
Re
Sc
Sh
St
Stm
2
k
ρcp
τs
ρu2∞ /2
α
DAB
hL
kf
Thermal diffusivity
Skin Friction Coefficient
Lewis Number - heat transfer vs. mass transport
Nusselt Number - Dimensionless Heat Transfer
P e = Rex P r
µCp
ν
=
α
k
ρu∞ x
u∞ x
=
µ
ν
ν
DAB
hm L
DAB
h
N uL
=
ρV cp
ReL P r
ShL
hm
=
V
ReL Sc
Peclet Number
Prandtl Number - momentum diffusivity vs. thermal diffusivity
Reynolds Number - Inertia vs. Viscosity
Schmidt Number momentum vs. mass transport
Sherwood Number - Dimensionless Mass Transfer
Stanton Number - Modified Nusselt Number
Stanton mass Number - Modified Sherwood Number
Boundary Layer Analogies - Require Geometric Similarity
Table 2: Boundary Layer Analogies
Sh
Nu
=
n
Pr
Scn
Heat and Mass Analogy
Applies always for same geometry, n is positive
hL
hm L
=
n
kP r
DAB Scn
Chilton Colburn Heat
jH =
Chilton Colburn Mass
jM =
Cf
= StP r2/3
2
0.6 < P r < 60
Cf
= Stm Sc2/3
2
0.6 < Sc < 3000
2/17
JRB, ASR
3
MEAM333 - Convection Correlations
External Flow
These typically use properties at the film temperature Tf =
3.1
Ts + T∞
2
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
Local Shear Stress
δ=p
5.0
u∞
p/vx
τs = 0.332u∞ ρµu∞ /x
Local Skin Friction Coefficient
Cf,x =
Local Heat Transfer
N ux =
Local Mass Transfer
Shx =
0.664Rex−0.5
hx x
= 0.332Rex0.5 P r1/3
k
hm,x x
1/3
= 0.332Re0.5
x Sc
DAB
Cf,x = 1.328Rex−0.5
Average Skin Friction Coefficient
Average Heat Transfer
N ux =
Average Mass Transfer
Shx =
hx x
= 0.664Rex0.5 P r1/3
k
hm,x x
1/3
= 0.664Re0.5
x Sc
DAB
N ux
N ux = 0.565P ex0.5
N ux
1/3
0.3387Re0.5
x Pr
N ux = 1/4
1 + (0.0468/P r)2/3
Re < 5E5
Re < 5E5
Re < 1
Re < 5E5
P r ≥ 0.6
Re < 5E5
Sc ≥ 0.6
Re < 1
Isothermal
Re < 5E5
P r ≥ 0.6
Re < 5E5
Sc ≥ 0.6
Liquid Metals
N ux = 2N ux
P r ≤ 0.05
P ex ≥ 100
All Prandtl Numbers
P ex ≥ 100
Table 4: Turbulent Flow Over an Isothermal Plate Rex > 5 · 105
Skin Friction Coefficient
Cf,x = 0.0592Rex−0.2
5E5 < Re < 108
Boundary Layer Thickness
δ = 0.37xRex−0.2
Heat Transfer
1/3
N ux = StRex P r = 0.0296Re0.8
x Pr
Mass Transfer
1/3
Shx = StRex Sc = 0.0296Re0.8
x Sc
5E5 < Re < 108
5E5 < Re < 108
0.6 < P r < 60
5E5 < Re < 108
0.6 < P r < 3000
3/17
JRB, ASR
3.2
MEAM333 - Convection Correlations
Mixed Flow Over a plate
If transition occurs at xLc ≥ 0.95 The laminar plate model may be used for h. Once the critical transition point
0.5
has been found, we define A = 0.037Re0.8
x,c − 0.664Rex,c These typically use properties at the film temperature
Ts + T∞
Tf =
2
Table 5: Mixed Flow Over an Isothermal Plate
CfL = 0.074Re−0.2 −
Average Skin Friction Coefficient
2A
ReL
5 · 105 < ReL < 108
0.6 < Sc < 60
5 · 105 < ReL < 108
1/3
ShL = (0.037Re0.8
L − A)Sc
Average Mass Transfer
3.3
0.6 < P r < 60
5 · 105 < ReL < 108
1/3
N uL = (0.037Re0.8
L − A)P r
Average Heat Transfer
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
Local Heat Transfer
Average Heat Transfer
3.4
N ux =
N ux |ζ=0
[1 −
1/3
(ζ/x)0.75 ]
N ux |ζ=0
N ux = 1/9
1 − (ζ/x)9/10
h
i
p+1 p/(p+1)
L
1 − (ζ/L) p+2
N uL = N uL |ζ=0 L−ζ
laminar
0 < ReL < 5 · 105
turbulent
5 · 105 < ReL < 108
p = 2 Laminar Flow
p = 8 Turbulent Flow
Plates with Constant Heat Flux
For average heat transfer values, it is acceptable to use the isothermal results for T =
R
0
L(Ts − T∞ )dx
Table 7: Constant Heat Flux
Local Heat Transfer Laminar
Local Heat Transfer Turbulent
3.5
1/3
N ux = 0.453Re0.5
x Pr
1/3
N ux = 0.0308Re0.8
x Pr
0 < ReL < 5 · 105
P r > 0.6
ReL > 5 · 105
0.6 < P r < 60
Cylinder in Cross Flow
For the cylinder in cross flow, we use ReD =
Ts + T∞
Tf =
2
ρV D
µ
=
VD
ν
These typically use properties at the film temperature
4/17
JRB, ASR
MEAM333 - Convection Correlations
Table 8: Cylinder in Cross Flow
N uD =
1/3
CRem
DP r
N uD =
n
CRem
DP r
N uD
3.6
Pr
P rs
0.7 < P r < 60
C, m are found as functions
of ReD on P426
0.7 < P r < 500
1 < ReD < 106
All properties evaluated at
T∞ except P rs
Uses table 7.4 P428
0.25
"
5/8 #4/5
1/3
Red
0.62Re0.5
D Pr
= 0.3 + 1/4 1 + 282, 000
1 + (0.4/P r)2/3
P r > 0.2
Flow over Spheres
Table 9: Flow over Spheres
2/3
0
N uD = 2 + (0.4Re0.5
D + 0.06ReD )P r .4
µ
µs
1/4
1/3
N uD = 2 + 0.6Re0.5
D Pr
0.71 < P r < 380
3.5 < P r < 6.6 · 104
1.0 < (µ/µs ) < 3.2
All properties except µs
are evaluated at T∞
For Freely Falling Drops
Infinite Stationary Medium
Red → 0
N uD = 2
5/17
JRB, ASR
3.7
3.7.1
MEAM333 - Convection Correlations
Flow Through Banks of Tubes
Geometric Properties
Table 10: Tube Bank Properties
ReD =
Vmax =
Vmax =
ρVmax D
µ
ST
Vi
ST − D
ST
Vi
2(SD − D)
Aligned OR
ST + D
2
ST + D
<
2
Staggered and SD >
Staggered and SD
Figure 1: Tube bank geometries for aligned (a) and staggered (b) banks
6/17
JRB, ASR
3.7.2
MEAM333 - Convection Correlations
Flow Correlations
Table 11: Flow through banks of tubes
1/3
N uD = 1.13C1 Rem
D,max P r
N uD |(NL <10) = C2 N uD |(NL ≥10)
N uD =
0.36
CRem
D,max P r
Pr
P rs
0.25
N uD |(NL <20) = C2 N uD |(NL ≥20)
More than 10 rows of tubes
2000 < ReD,max < 40, 000
P r > 0.7
Coefficients come from
table 7.5 on P438
C2 comes from Table 7.6 on P439
2000 < ReD,max < 40, 000
P r > 0.7
Coefficients come from
table 7.5 on P438
C, m comes from Table 7.7 on P440
1000 < ReD,max < 2 · 106
0.7 < P r < 500
More than 20 rows
For the above correlation
C2 comes from Table 7.8 on P440
2000 < ReD,max < 40, 000
P r > 0.7
Table 12: Flow through banks of tubes 2
(Ts − Ti ) − (Ts − T o)
−Ti
ln TTss−T
o
Ts − To
πDN h̄
Dimensionless Temp Correlation
= exp −
Ts − Ti
ρV NT ST cP
N - total number of tubes, NT - total number of tubes in transverse plane
Heating Per Unit Length
q 0 = N h̄πD∆Tlm
Log Mean Temp.
∆Tlm =
7/17
JRB, ASR
3.8
MEAM333 - Convection Correlations
Impinging Jets
Heat and mass transfer is measured against the fluid properties at the nozzle exit q 00 = h(Ts − Te ) The Reynolds
A
and Nusselt numbers are measured using the hydraulic diameter of the nozzle Dh = Pc,e The Reynolds number
uses the nozzle exit velocity. All correlations use the target cell region Ar which is affected by the nozzle. This is
depicted in Figure 7.17 on P449. H is the height from the plate to the nozzle exit
Table 13: Impinging Jets
Single
Round Nozzle
G factor
Round Nozzle
Array
K factor
Single
Slot Nozzle
N u = P r0.42 G Ar , H
2Re0.5 (1 + 0.005Re0.55 )0.5
D
G = 2A0.5
r
N u = P r0.42 0.5K Ar ,
Slot Nozzle
Array
Ar,o
H
D
G Ar ,
H
D
Always
Re2/3
6 −0.05
H/D
K = 1 + 0.6/Ar1/2
Nu = Pr
0.42
"
m factor
1 − 2.2A0.5
r
1 + 0.2(H/d − 6)Ar0.5
2 3/4
N u = P r0.42 Ar,o
3
m = 0.695 −
+
H
2W
H
2W
−2
8/17
3000 < Re < 9 · 104
2 < H/D < 10
0.025 < Ar < 0.125
#−1
1.33
+ 3.06
2Re
Ar /Ar,o + Ar,o /Ar
h
Ar,o = 60 + 4
2000 < Re < 105
2 < H/D < 12
0.004 < Ar < 0.04
Always
3.06
Rem
0.5/Ar + H/W + 2.78
1
4Ar
2000 < Re < 4 · 105
2 < H/D < 12
0.004 < Ar < 0.04
2 i−0.5
2/3
Always
SH
WL
≥1
1500 < Re < 4 · 104
2 < H/D < 80
0.008 < Ar < 2.5Ar,o
Always
JRB, ASR
3.9
MEAM333 - Convection Correlations
Packed Beds
For packed beds, the heat transfer depends on the total particle surface area Ap,t
q = h̄Ap,t ∆Tlm
The outlet temperature can be determined from the log mean relation
Ts − To
h̄Ap,t
= exp −
Ts − Ti
ρVi Ac,b cp
For Spheres :
−0.575
j̄H = j̄m = 2.06ReD
where Pr or Sc ≈ 0.7 and 90 < ReD < 4000 For non spheres multiply the right hand side by a factor - uniform
cylinders of L = D use 0.71, for uniform cubes use 0.71
is the porosity and is typically 0.3 to 0.5.
4
4.1
4.1.1
Internal Flow
Circular Tube
Properties
Table 14: Flow Conditions
Mean Velocity
ReD
Hydrodynamic Entry Length
Velocity Profile
um =
ṁ
ρAc
ρum D
µm D
=
µ
ν
x
f d,h
≈ 0.05ReD
D lam
x
f d,h
≤ 60
10 ≤
D turb
"
2 #
u(r)
r
=2 1−
um
r0
ReD ≡
f≡
−(dp/dx)D
ρu2m /s
f=
64
ReD
Moody Friction Factor
−1/4
f = 0.316ReD
−1/4
f = 0.184ReD
f = (0.790ln(ReD ) − 1.64)−2
Power for Pressure Drop
P = (∆p)∀˙
9/17
turbulent onset @ ReD ≈ 2300
Smooth
ReD ≤ 2 × 104
Smooth
ReD ≥ 2 × 104
Smooth
3000 ≤ ReD ≤ 5 × 106
ṁ
∀˙ =
ρ
JRB, ASR
MEAM333 - Convection Correlations
Table 15: Constant Surface Heat Flux
Convective Heat Transfer
Mean Temperature
qconv = qs00 (P L)
q 00 P
Tm (x) = Tm,i + s x
ṁcp
qs00 = constant
qs00 = constant
Table 16: Constant Surface Temperature
Convective Heat Transfer
∆Tlm
Log Mean Temperature
qconv = hAs ∆Tlm
∆To − ∆Ti
≡
ln(∆To /∆Ti )
Ts = constant
Ts = constant
∆To
Ts − Tm (x)
P xh
=
= exp −
∆Ti
Ts − Tm,i
ṁcp
Table 17: Constant External Environment Temperature
Heat Transfer
q = U As ∆Tlm
T∞ = constant
Log Mean Temperature
4.1.2
∆To
U As
T∞ − Tm (x)
=
= exp −
∆Ti
T∞ − Tm,i
ṁcp
Flow Correlations
Table 18: Fully Developed Flow In Circular Tubes
lamniar
N uD
hD
≡
= 4.36
k
fully developed
qs00 = constant
lamniar
N uD
hD
≡
= 3.66
k
fully developed
Ts = constant
10/17
T∞ = constant
JRB, ASR
MEAM333 - Convection Correlations
Table 19: Laminar Entry Region Flow In Circular Tubes
N uD ≡
hD
0.0668(D/L)ReD P r
= 3.66 +
k
1 + 0.04[(D/L)ReD P r]2/3
N uD ≡
hD
= 1.86
k
ReD P r
L/D
1/3 µ
µs
0.14
lamniar
Ts = constant
(thermal entry length)
OR
(combined with Pr ≥ 5)
lamniar
Ts = constant
0.60 ≤ P r≤ 5
µ
0.0044 ≤
≤ 9.75
µs
All properties evaluated at the mean temperature Tm = (Tm,i + Tm,o )/2
Table 20: Turbulent Flow In Circular Tubes
N uD
N uD
hD
4/5
≡
= 0.023ReD P rn
k
Ts > Tm : n = 0.4
Ts < Tm : n = 0.3
hD
4/5
≡
= 0.027ReD P r1/3
k
µ
µs
0.14
turbulent
fully developed
small temperature diff
0.6 ≤ P r ≤ 160
ReD ≥ 10, 000
laminar
0.7 ≤ P r ≤ 16, 700
ReD ≥ 10, 000
L
≥ 10
D
lamniar
N uD
0.5 ≤ P r ≤ 2000
3000 ≤ ReD ≤ 5 × 106
Above appropriate for both constant Ts and constant qs00
lamniar
NOT liquid metals (3 × 10−3 ≤ P r ≤ 5 × 10−2 )
hD
0.827
N uD ≡
= 4.82 + 0.0185P eD
qs00 = constant
k
3.6 × 103 ≤ ReD ≤ 9.05 × 105
102 ≤ P eD ≤ 104
similarly as immediately above
hD
N uD ≡
= 5.0 + 0.025P e0.8
Ts = constant
D
k
100 ≤ P eD
All properties evaluated at the mean temperature Tm = (Tm,i + Tm,o )/2
hD
(f /8)(ReD − 1000)P r
≡
=
k
1 + 12.7(f /8)1/2 (P r2/3 − 1)
11/17
JRB, ASR
4.2
4.2.1
MEAM333 - Convection Correlations
Non-Circular Tubes
Properties
Table 21: Flow in Non-Circular Tubes
Hydrodynamic Diameter
ReDh
ReDh
4Ac
Dh ≡
P
ρum Dh
µm Dh
≡
=
µ
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
4.3
4.3.1
MEAM333 - Convection Correlations
Concentric Tube Annulus
Properties
Table 22: Concentric Tube Annulus Properties
Interior heat transfer
Exterior heat transfer
Hydrodynamic Diameter
4.3.2
qi00 = hi (Ts,i − Tm )
qo00 = ho (Ts,o − Tm )
Dh = Do − Di
Flow Correlations
Table 23: Correlations for Concentric Tube Annulus
lamniar
fully developed
one surface insulated
one surface const Ts
See Table 8.2 on Page 520
N ui =
N uii
N uoo
, N uo =
1 − (qo00 /qi00 )θi∗
1 − (qi00 /qo00 )θo∗
See Table 8.3 for above parameters as a function of
4.4
Di
Do
laminar
qi00 = constant
qo00 = constant
Heat Transfer Enhancement - Tube Coiling
Table 24: Properties for Helically Coiled Tubes
D,C are defined
in Figure 8.13
on Page 522
ReD,c,h = ReD,c [1 + 12(D/C)0.5 ]
ReD,c = 2300
Critical
Reynolds Number
f
f=
f
f=
f
f
64
ReD
ReD (D/C)1/2 ≤ 30
27
(D/C)0.1375
0
ReD .725
7.2
=
(D/C)0.25
Re0D .5
30 ≤ ReD (D/C)1/2 ≤ 300
300 ≤ ReD (D/C)1/2
Table 25: Correlations for Helically Coiled Tubes
"
N uD =
4.343
3.66 +
a
3
+ 1.158
a=
ReD (D/C)1/2
b
927(C/D)
1+
Re2D P r
b=1+
0.477
Pr
13/17
3/2 #1/3 µ
µs
0.14
0.005 ≤ P r ≤ 1600
1 ≤ ReD D
C
1/2
≤ 1000
JRB, ASR
4.5
MEAM333 - Convection Correlations
Internal Convection Mass Transfer
Table 26: Properties for Internal Convection Mass Transfer
R
Mean
Species Density
Mean
Species Density
Local
Mass Flux
ρA,m =
ρA,m =
Ac
2
um ro2
(ρA u)dAc
u m Ac
R ro
(ρA ur)dr
0
Any Shape
Circular Tube
n00A = hm (ρA,s − ρA,m )
nA = hm As ∆ρA,lm
Total
Mass Flux
nA =
∆ρA,lm =
Log Mean
Concentration Difference
ṁ
(ρA,o − ρA, i)
ρ
∆ρA,o − ∆ρA,i
ln(∆ρA,o /∆ρA,i )
∆ρA (x)
ρA,s − ρA,m (x)
hm ρP
x
=
= exp −
∆ρA,i
ρA,s − ρA,m,i
ṁ
ShD =
hm D
DA B
ShD =
hm D
DA B
Sherwood Number
The concentration entry length xf d,c can be determined with the mass transfer analogy and the same function
used to determine xf d,t . From this point, the appropriate heat transfer correlation can be invoked along the lines
of the mass transfer analogy,
5
Natural Convection
Natural Convection uses the Rayleigh number instead of the Reynolds number. Transition to turbulent flow
happens around
Ra ≈ 109
14/17
JRB, ASR
5.1
MEAM333 - Convection Correlations
Natural Convection, Vertical Plate
Table 27: Natural Convection, Vertical Plate
Laminar Heat Transfer
N ux =
g factor
g(P r) =
Better avg. Heat Transfer
1/4
uses g below
g(P r)
0.75P r0.5
(0.609 + 1.221P r0.5 + 1.238P r)1/4
4
N uL =
3
"
Average Laminar
Grx
4
N uL = 0.825 + Grx
4
0 < Pr < ∞
1/4
g(P r)
laminar
#2
1/6
0.387Ral
1 + (0.492/P r)9/16
8/27
Applies for all RaL
1/4
0.670Ral
N uL = 0.68 + 4/9
1 + (0.492/P r)9/16
Better avg. Laminar Heat Transfer
5.2
RaL < 109
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 are
recommended 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 Plate
Lower Surface Cold Plate
Upper Surface Hot Plate
Lower Surface Cold Plate
Lower Surface Hot Plate
Upper Surface Cold Plate
5.4
1/4
104 < RaL < 107
1/3
107 < RaL < 101 1
1/4
105 < RaL < 101 0
N uL = 0.54RaL
N uL = 0.15RaL
N uL = 0.27RaL
Long Horizontal Cylinder
Assumes isothermal cylinder. The following correlation applies for RaD < 101 2
"
1/6
0.387RaD
N uD = 0.60 + 8/27
1 + (0.559/P r)9/16
5.5
Spheres
For P r > 0.7 and RaD < 101 1
1/4
0.589RaD
N uD = 2 + 4/9
1 + (0.469/P r)9/16
15/17
#2
JRB, ASR
5.6
MEAM333 - Convection Correlations
Vertical Channels
This section describes correlations for natural convection between to parralel plates. It uses Ras which uses the
plate separation for the length scale. I believe that the convection area is the surface area where heating/cooling
happens.
Table 29: Vertical Channels
Symmetrically Heated
Isothermal Plates
Symmetrically Heated
Isothermal Plates
1 Insulated Plate
2 Isothermal Plate
Isothermal /
Adiabatic
(Better)
N us =
1
24 Ras
0.75
S
35
1 − exp −
L
Ras (S/L)
N us =
RAs (S/L)
24
N us =
Ras (S/L)
12
10−1 <
S
L Ras
10−1 <
S
L →0
10−1 <
S
L →0
< 105
S
L Ras
< 105
S
L Ras
< 105
−1/2
C1
C2
S
N us =
+
Ras L
≤ 10
(Ras S/L)2
(Ras S/L)1/2
q/A
S
gβ(Ts − T∞ )S 3
The isothermal correlations use N us =
and Ras =
Ts − T∞ k
αν
The better isothermal correlation uses
C1 = 576, C2 = 2.87 for Symmetric isothermal Plates
C1 = 144, C2 = 2.87 for isothermal and adiabatic Plates
Symmetric
0.5
N us,L,f d = 0.144 [Ra∗s (S/L)]
Uses Ra∗
Isoflux Plates
1 Isoflux Plate
0.5
N us,L,f d = 0.204 [Ra∗s (S/L)]
Uses Ra∗
1 Insulated
−1/2
Isoflux /
C1
C2
S
N us,L =
+
Adiabatic
Ras L
≥ 100
Ra∗s S/L (Ra∗s S/L)2/5
(Better)
gβqs00 S 4
qs00
S
and Ra∗s =
The isoflux corelations use N us,f d =
Ts,L − T∞ k
kαν
The better isoflux correlation uses
C1 = 48, C2 = 2.51 for Symmetric isoflux Plates
C1 = 24, C2 = 2.51 for isoflux and adiabatic Plates
5.7
Inclined Channels
For plates inclined less than 45 degrees from the vertical
1/4
N us = 0.645 [Ras (S/L)]
Fluid properties are evaluated at T̄ =
5.8
Ts +T∞
2
This requires Ras (S/L) > 200
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 heat
transferring plates. Inclined plates are discussed on P590.
16/17
JRB, ASR
MEAM333 - Convection Correlations
Table 30: Rectangular Channels
Horizontal Cavity
Heated from Below
Heat transfer on
Vertical Surfaces
1/3
N uL = 0.069RaL P r0.074
N uL = 0.22
Heat transfer on
Vertical Surfaces
Heat transfer on
Vertical Surfaces
N uL = 0.18
N uL =
Heat transfer on
Vertical Surfaces
5.9
Pr
RaL
0.2 + P r
0.28 Pr
RaL
0.2 + P r
0.012
0.42Ra0.25
L Pr
H
L
−0.25
0.29
H
L
−0.3
1/3
N uL = 0.046RaL
Concentric Cylinders
For Cylinders we use an effective thermal conductivity
kef f
= 0.386
k
Pr
0.861 + P r
1/4
Ra1/4
c
The Rayleigh number uses the corrected length
4/3
Lc =
2 [ln(ro /ri )]
(ri−0.6 + ro−0.6 )5/3
q=
2πLkef f (Ti − To )
ln(ro /ri )
The Heat Transfer is found as
5.10
Concentric Spheres
For Spheres we use an effective thermal conductivity
kef f
= 0.74
k
Pr
0.861 + P r
1/4
Ra1/4
s
The Rayleigh number uses the corrected length
Ls =
1
ri
−7/5
21/3 (ri
The Heat Transfer is found as
q=
1
ro
−
4/3
−7/5 5/3
)
+ ro
4πLkef f (Ti − To )
(1/ri ) − (1/ro )
17/17
3 · 105 < RaL < 7 · 109
All properties evaluated at
average temp. between
hot and cold plates
103 < RaL < 101 0
2≤ H
L ≤ 10
P r ≤ 105
RaL P r
103 < 0.2+P
r
1≤ H
≤
2
L
10−3 ≤ P r ≤ 105
104 < RaL < 107
10 ≤ H
L ≤ 40
1 ≤ P r ≤ 2 · 104
106 < RaL ≤ 109
1≤ H
L ≤ 40
1 ≤ P r ≤ 20