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 3.1 External Flow for a Flat Plate ...... 3 3.2 Mixed Flow Over a plate ...... 4 3.3 Unheated Starting Length ...... 4 3.4 Plates with Constant Heat Flux ...... 4 3.5 Cylinder in Cross Flow ...... 4 3.6 Flow over Spheres ...... 5 3.7 Flow Through Banks of Tubes ...... 6 3.7.1 Geometric Properties ...... 6 3.7.2 Flow Correlations ...... 7 3.8 Impinging Jets ...... 8 3.9 Packed Beds ...... 9

4 Internal Flow 9 4.1 Circular Tube ...... 9 4.1.1 Properties ...... 9 4.1.2 Flow Correlations ...... 10 4.2 Non-Circular Tubes ...... 12 4.2.1 Properties ...... 12 4.2.2 Flow Correlations ...... 12 4.3 Concentric Tube Annulus ...... 13 4.3.1 Properties ...... 13 4.3.2 Flow Correlations ...... 13 4.4 Heat Transfer Enhancement - Tube Coiling ...... 13 4.5 Internal Convection Mass Transfer ...... 14

5 Natural Convection 14 5.1 Natural Convection, Vertical Plate ...... 15 5.2 Natural Convection, Inclined Plate ...... 15 5.3 Natural Convection, Horizontal Plate ...... 15 5.4 Long Horizontal Cylinder ...... 15 5.5 Spheres ...... 15 5.6 Vertical Channels ...... 16 5.7 Inclined Channels ...... 16 5.8 Rectangular Cavities ...... 16 5.9 Concentric Cylinders ...... 17 5.10 Concentric Spheres ...... 17

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1 Dimensionless Parameters

Table 1: Dimensionless Parameters

k α Thermal diﬀusivity ρcp τs Cf 2 Skin Friction Coeﬃcient ρu∞/2 α Le Lewis Number - heat transfer vs. mass transport DAB hL Nu Nusselt Number - Dimensionless Heat Transfer kf

P e P e = RexP r Peclet Number ν µC P r = p Prandtl Number - momentum diffusivity vs. thermal diffusivity α k ρu x u x Re ∞ = ∞ Reynolds Number - Inertia vs. Viscosity µ ν ν Sc Schmidt Number momentum vs. mass transport DAB h L Sh m Sherwood Number - Dimensionless Mass Transfer DAB h Nu St = L Stanton Number - Modified Nusselt Number ρV cp ReLP r hm ShL Stm = Stanton mass Number - Modified Sherwood Number V ReLSc

2 Boundary Layer Analogies - Require Geometric Similarity

Table 2: Boundary Layer Analogies

Nu Sh = P rn Scn Heat and Mass Analogy Applies always for same geometry, n is positive hL hmL n = n kP r DABSc C j = f = StP r2/3 Chilton Colburn Heat H 2 0.6 < P r < 60 C j = f = St Sc2/3 Chilton Colburn Mass M 2 m 0.6 < Sc < 3000

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3 External Flow T + T These typically use properties at the ﬁlm temperature T = s ∞ f 2 3.1 External Flow for a Flat Plate T + T These use properties at the ﬁlm temperature T = s ∞ f 2

Table 3: Flat Plate Isothermal Laminar Flow 5.0 Flat plate Boundary Layer Thickness δ = p Re < 5E5 u∞/vx p Local Shear Stress τs = 0.332u∞ ρµu∞/x Re < 5E5 −0.5 Local Skin Friction Coeﬃcient Cf,x = 0.664Rex Re < 1

hxx 0.5 1/3 Re < 5E5 Local Heat Transfer Nux = = 0.332Re P r k x P r ≥ 0.6

hm,xx 0.5 1/3 Re < 5E5 Local Mass Transfer Shx = = 0.332Rex Sc DAB Sc ≥ 0.6 −0.5 Average Skin Friction Coeﬃcient Cf,x = 1.328Rex Re < 1 Isothermal h x Average Heat Transfer Nu = x = 0.664Re0.5P r1/3 Re < 5E5 x k x P r ≥ 0.6

hm,xx 0.5 1/3 Re < 5E5 Average Mass Transfer Shx = = 0.664Rex Sc DAB Sc ≥ 0.6 Liquid Metals

0.5 Nux = 2Nux Nux Nux = 0.565P e x P r ≤ 0.05

P ex ≥ 100 0.3387Re0.5P r1/3 All Prandtl Numbers Nu Nu = x x x 1/4 1 + (0.0468/P r)2/3 P ex ≥ 100

5 Table 4: Turbulent Flow Over an Isothermal Plate Rex > 5 · 10

−0.2 8 Skin Friction Coeﬃcient Cf,x = 0.0592Rex 5E5 < Re < 10 −0.2 8 Boundary Layer Thickness δ = 0.37xRex 5E5 < Re < 10 8 0.8 1/3 5E5 < Re < 10 Heat Transfer Nux = StRexP r = 0.0296Re P r x 0.6 < P r < 60 8 0.8 1/3 5E5 < Re < 10 Mass Transfer Shx = StRexSc = 0.0296Re Sc x 0.6 < P r < 3000

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3.2 Mixed Flow Over a plate

xc If transition occurs at L ≥ 0.95 The laminar plate model may be used for h. Once the critical transition point 0.8 0.5 has been found, we deﬁne A = 0.037Rex,c − 0.664Rex,c These typically use properties at the ﬁlm temperature T + T T = s ∞ f 2

Table 5: Mixed Flow Over an Isothermal Plate

0.8 1/3 0.6 < P r < 60 Average Heat Transfer NuL = (0.037ReL − A)P r 5 8 5 · 10 < ReL < 10 −0.2 2A 5 8 Average Skin Friction Coeﬃcient CfL = 0.074Re − 5 · 10 < ReL < 10 ReL 0.8 1/3 0.6 < Sc < 60 Average Mass Transfer ShL = (0.037ReL − A)Sc 5 8 5 · 10 < ReL < 10

3.3 Unheated Starting Length T + T Here the plate has T = T until x = ζ These typically use properties at the ﬁlm temperature T = s ∞ s ∞ f 2

Table 6: Unheated Starting Length

Nu | laminar Local Heat Transfer Nu = x ζ=0 x 1/3 5 [1 − (ζ/x)0.75] 0 < ReL < 5 · 10 Nu | turbulent Local Heat Transfer Nu = x ζ=0 x 1/9 5 8 1 − (ζ/x)9/10 5 · 10 < ReL < 10 p/(p+1) L h p+1 i p = 2 Laminar Flow Average Heat Transfer NuL = NuL|ζ=0 1 − (ζ/L) p+2 L−ζ p = 8 Turbulent Flow

3.4 Plates with Constant Heat Flux R For average heat transfer values, it is acceptable to use the isothermal results for T = 0 L(Ts − T∞)dx

Table 7: Constant Heat Flux

5 0.5 1/3 0 < ReL < 5 · 10 Local Heat Transfer Laminar Nux = 0.453Re P r x P r > 0.6 5 0.8 1/3 ReL > 5 · 10 Local Heat Transfer Turbulent Nux = 0.0308Re P r x 0.6 < P r < 60

3.5 Cylinder in Cross Flow ρV D VD For the cylinder in cross ﬂow, we use ReD = µ = ν These typically use properties at the ﬁlm temperature T + T T = s ∞ f 2

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Table 8: Cylinder in Cross Flow

0.7 < P r < 60 m 1/3 NuD = CReD P r C, m are found as functions of ReD on P426 0.7 < P r < 500 6 0.25 1 < ReD < 10 m n P r NuD = CReD P r All properties evaluated at P rs T∞ except P rs Uses table 7.4 P428 " #4/5 0.62Re0.5P r1/3 Re 5/8 Nu = 0.3 + D 1 + d D 1/4 P r > 0.2 1 + (0.4/P r)2/3 282, 000

3.6 Flow over Spheres

Table 9: Flow over Spheres

0.71 < P r < 380 4 1/4 3.5 < P r < 6.6 · 10 0.5 2/3 0 µ NuD = 2 + (0.4ReD + 0.06ReD )P r .4 1.0 < (µ/µs) < 3.2 µs All properties except µs

are evaluated at T∞ 0.5 1/3 NuD = 2 + 0.6ReD P r For Freely Falling Drops Inﬁnite Stationary Medium NuD = 2 Red → 0

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3.7 Flow Through Banks of Tubes 3.7.1 Geometric Properties

Table 10: Tube Bank Properties

ρV D Re = max D µ Aligned OR ST Vmax = Vi ST + D ST − D Staggered and S > D 2 ST ST + D Vmax = Vi Staggered and SD < 2(SD − D) 2

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

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3.7.2 Flow Correlations

Table 11: Flow through banks of tubes

More than 10 rows of tubes

2000 < ReD,max < 40, 000 m 1/3 NuD = 1.13C1ReD,maxP r 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

NuD|(NL<10) = C2NuD|(NL≥10) P r > 0.7 Coeﬃcients come from table 7.5 on P438 C, m comes from Table 7.7 on P440 0.25 6 m 0.36 P r 1000 < ReD,max < 2 · 10 NuD = CReD,maxP r P rs 0.7 < P r < 500 More than 20 rows For the above correlation

C2 comes from Table 7.8 on P440 NuD|(NL<20) = C2NuD|(NL≥20) 2000 < ReD,max < 40, 000 P r > 0.7

Table 12: Flow through banks of tubes 2

(Ts − Ti) − (Ts − T o) Log Mean Temp. ∆Tlm = ln Ts−Ti Ts−To T − T πDNh¯ Dimensionless Temp Correlation s o = exp − Ts − Ti ρV NT ST cP N - total number of tubes, NT - total number of tubes in transverse plane 0 ¯ Heating Per Unit Length q = NhπD∆Tlm

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3.8 Impinging Jets 00 Heat and mass transfer is measured against the ﬂuid properties at the nozzle exit q = h(Ts − Te) The Reynolds Ac,e and Nusselt numbers are measured using the hydraulic diameter of the nozzle Dh = P 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

2000 < Re < 4 · 105 Single 0.42 H 0.5 0.55 0.5 Nu = P r G Ar, 2Re (1 + 0.005Re ) 2 < H/D < 12 Round Nozzle D 0.004 < Ar < 0.04 1 − 2.2A0.5 G factor G = 2A0.5 r Always r 1 + 0.2(H/d − 6)Ar0.5 2000 < Re < 105 Round Nozzle 0.42 H H 2/3 Nu = P r 0.5K Ar, G Ar, Re 2 < H/D < 12 Array D D 0.004 < Ar < 0.04 6−0.05 H/D K factor K = 1 + 0.6/Ar1/2 Always 3000 < Re < 9 · 104 Single 3.06 Nu = P r0.42 Rem 2 < H/D < 10 Slot Nozzle 0.5/Ar + H/W + 2.78 0.025 < Ar < 0.125 " #−1 1 H 1.33 m factor m = 0.695 − + + 3.06 Always 4Ar 2W SH WL ≥ 1 2/3 4 Slot Nozzle 0.42 2 3/4 2Re 1500 < Re < 4 · 10 Nu = P r Ar,o Array 3 Ar/Ar,o + Ar,o/Ar 2 < H/D < 80

0.008 < Ar < 2.5Ar,o −0.5 h H 2i Ar,o Ar,o = 60 + 4 2W − 2 Always

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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

T − T hA¯ s o = exp − p,t Ts − Ti ρViAc,bcp For Spheres: ¯ ¯ −0.575 jH = jm = 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 Internal Flow 4.1 Circular Tube 4.1.1 Properties

Table 14: Flow Conditions

m˙ um = Mean Velocity ρAc

ρumD µmD ReD ≡ = ReD µ ν turbulent onset @ ReD ≈ 2300

xfd,h ≈ 0.05ReD D lam Hydrodynamic Entry Length x 10 ≤ fd,h ≤ 60 D turb " # u(r) r 2 = 2 1 − Velocity Proﬁle um r0

−(dp/dx)D f ≡ 2 ρum/s Moody Friction Factor 64 f = ReD

−1/4 Smooth f = 0.316ReD 4 ReD ≤ 2 × 10 −1/4 Smooth f = 0.184ReD 4 ReD ≥ 2 × 10 −2 Smooth f = (0.790ln(ReD) − 1.64) 6 3000 ≤ ReD ≤ 5 × 10 m˙ Power for Pressure Drop P = (∆p)∀˙ ∀˙ = ρ

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Table 15: Constant Surface Heat Flux

00 00 Convective Heat Transfer qconv = qs (PL) qs = constant q00P s 00 Mean Temperature Tm(x) = Tm,i + x qs = constant mc˙ p

Table 16: Constant Surface Temperature

Convective Heat Transfer qconv = hAs∆Tlm Ts = constant ∆To − ∆Ti ∆Tlm ≡ ln(∆To/∆Ti) Log Mean Temperature ∆T T − T (x) P xh Ts = constant o = s m = exp − ∆Ti Ts − Tm,i mc˙ p

Table 17: Constant External Environment Temperature

Heat Transfer q = UAs∆Tlm T∞ = constant ∆To T∞ − Tm(x) UAs Log Mean Temperature = = exp − T∞ = constant ∆Ti T∞ − Tm,i mc˙ p

4.1.2 Flow Correlations

Table 18: Fully Developed Flow In Circular Tubes

lamniar hD NuD ≡ = 4.36 fully developed k 00 qs = constant lamniar hD NuD ≡ = 3.66 fully developed k Ts = constant

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Table 19: Laminar Entry Region Flow In Circular Tubes

lamniar T = constant hD 0.0668(D/L)Re P r s Nu ≡ = 3.66 + D D 2/3 (thermal entry length) k 1 + 0.04[(D/L)ReDP r] OR (combined with Pr ≥ 5) lamniar

Ts = constant 1/3 0.14 hD ReDP r µ 0.60 ≤ P r ≤ 5 NuD ≡ = 1.86 k L/D µs 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

turbulent hD Nu ≡ = 0.023Re4/5P rn fully developed D k D small temperature diﬀ Ts > Tm : n = 0.4 0.6 ≤ P r ≤ 160 Ts < Tm : n = 0.3 ReD ≥ 10, 000 laminar 0.7 ≤ P r ≤ 16, 700 hD µ 0.14 4/5 1/3 Re ≥ 10, 000 NuD ≡ = 0.027ReD P r D k µs L ≥ 10 D

lamniar hD (f/8)(Re − 1000)P r Nu ≡ = D 0.5 ≤ P r ≤ 2000 D k 1 + 12.7(f/8)1/2(P r2/3 − 1) 6 3000 ≤ ReD ≤ 5 × 10 00 Above appropriate for both constant Ts and constant qs lamniar NOT liquid metals (3 × 10−3 ≤ P r ≤ 5 × 10−2) hD Nu ≡ = 4.82 + 0.0185P e0.827 q00 = constant D k D s 3 5 3.6 × 10 ≤ ReD ≤ 9.05 × 10 2 4 10 ≤ P eD ≤ 10 similarly as immediately above hD Nu ≡ = 5.0 + 0.025P e0.8 T = constant D k D s 100 ≤ P eD

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

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4.2 Non-Circular Tubes 4.2.1 Properties

Table 21: Flow in Non-Circular Tubes

4A Hydrodynamic Diameter D ≡ c h P ρumDh µmDh ReDh ≡ = ReDh 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

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4.3 Concentric Tube Annulus 4.3.1 Properties

Table 22: Concentric Tube Annulus Properties

00 Interior heat transfer qi = hi(Ts,i − Tm) 00 Exterior heat transfer qo = ho(Ts,o − Tm) Hydrodynamic Diameter Dh = Do − Di

4.3.2 Flow Correlations

Table 23: Correlations for Concentric Tube Annulus

lamniar fully developed See Table 8.2 on Page 520 one surface insulated

one surface const Ts Nuii Nuoo Nui = 00 00 ∗ , Nuo = 00 00 ∗ 1 − (qo /qi )θi 1 − (qi /qo )θo laminar 00 qi = constant See Table 8.3 for above parameters as a function of Di 00 Do qo = constant

4.4 Heat Transfer Enhancement - Tube Coiling

Table 24: Properties for Helically Coiled Tubes

D,C are deﬁned Critical Re = Re [1 + 12(D/C)0.5] D,c,h D,c in Figure 8.13 Reynolds Number Re = 2300 D,c on Page 522

64 1/2 f f = ReD(D/C) ≤ 30 ReD 27 0.1375 1/2 f f = 0 (D/C) 30 ≤ ReD(D/C) ≤ 300 ReD.725 7.2 0.25 1/2 f f = 0 (D/C) 300 ≤ ReD(D/C) ReD.5

Table 25: Correlations for Helically Coiled Tubes

1/3 " 3 1/2 3/2# 0.14 4.343 ReD(D/C) µ NuD = 3.66 + + 1.158 a b µs

927(C/D) 0.005 ≤ P r ≤ 1600 a = 1 + Re2 P r D 1/2 D 1 ≤ ReD C ≤ 1000 0.477 b = 1 + P r

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4.5 Internal Convection Mass Transfer

Table 26: Properties for Internal Convection Mass Transfer

R Mean (ρAu)dAc Ac ρA,m = Any Shape Species Density umAc

Mean 2 R ro ρA,m = 2 0 (ρAur)dr Circular Tube Species Density umro Local n00 = h (ρ − ρ ) Mass Flux A m A,s A,m

nA = hmAs∆ρA,lm Total m˙ Mass Flux n = (ρ − ρA, i) A ρ A,o

∆ρA,o − ∆ρA,i ∆ρA,lm = ln(∆ρA,o/∆ρA,i) Log Mean ∆ρ (x) ρ − ρ (x) h ρP Concentration Diﬀerence A = A,s A,m = exp − m x ∆ρA,i ρA,s − ρA,m,i m˙

hmD ShD = DAB Sherwood Number hmD ShD = DAB

The concentration entry length xfd,c can be determined with the mass transfer analogy and the same function used to determine xfd,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

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5.1 Natural Convection, Vertical Plate

Table 27: Natural Convection, Vertical Plate

Gr 1/4 Laminar Heat Transfer Nu = x g(P r) uses g below x 4

0.75P r0.5 g factor g(P r) = 0 < P r < ∞ (0.609 + 1.221P r0.5 + 1.238P r)1/4

4 Gr 1/4 Average Laminar Nu = x g(P r) laminar L 3 4 " #2 0.387Ra1/6 Better avg. Heat Transfer Nu = 0.825 + l Applies for all Ra L 8/27 L 1 + (0.492/P r)9/16

0.670Ra1/4 Better avg. Laminar Heat Transfer Nu = 0.68 + l Ra < 109 L 4/9 L 1 + (0.492/P r)9/16

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 are recommended for the other cases.

5.3 Natural Convection, Horizontal Plate

As These correlations use L = P

Table 28: Natural Convection, Horizontal Plate

Upper Surface Hot Plate 1/4 4 7 NuL = 0.54Ra 10 < RaL < 10 Lower Surface Cold Plate L

Upper Surface Hot Plate 1/3 7 1 NuL = 0.15Ra 10 < RaL < 10 1 Lower Surface Cold Plate L

Lower Surface Hot Plate 1/4 5 1 NuL = 0.27Ra 10 < RaL < 10 0 Upper Surface Cold Plate L

5.4 Long Horizontal Cylinder 1 Assumes isothermal cylinder. The following correlation applies for RaD < 10 2 " #2 0.387Ra1/6 Nu = 0.60 + D D 8/27 1 + (0.559/P r)9/16

5.5 Spheres 1 For P r > 0.7 and RaD < 10 1 0.589Ra1/4 Nu = 2 + D D 4/9 1 + (0.469/P r)9/16

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5.6 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

0.75 Symmetrically Heated 1 S 35 −1 S 5 Nus = 24 Ras 1 − exp − 10 < L Ras < 10 Isothermal Plates L Ras(S/L) Symmetrically Heated RA (S/L) 10−1 < S Ra < 105 Nu = s L s s 24 S Isothermal Plates L → 0 1 Insulated Plate Ra (S/L) 10−1 < S Ra < 105 Nu = s L s s 12 S 2 Isothermal Plate L → 0 Isothermal / C C −1/2 Nu = 1 + 2 S Adiabatic s 2 1/2 Ras L ≤ 10 (RasS/L) (RasS/L) (Better) 3 q/A S gβ(Ts − T∞)S The isothermal correlations use Nus = 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 ∗ Nus,L,fd = 0.144 [Ra (S/L)] Uses Ra Isoﬂux Plates s

1 Isoflux Plate ∗ 0.5 ∗ Nus,L,fd = 0.204 [Ra (S/L)] Uses Ra 1 Insulated s Isoflux / C C −1/2 Adiabatic Nu = 1 + 2 Ra S ≥ 100 s,L Ra∗S/L (Ra∗S/L)2/5 s L (Better) s s 00 00 4 qs S ∗ gβqs S The isoflux corelations use Nus,fd = and Ras = 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 Nus = 0.645 [Ras(S/L)]

¯ Ts+T∞ Fluid properties are evaluated at 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 heat transferring plates. Inclined plates are discussed on P590.

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Table 30: Rectangular Channels

5 9 3 · 10 < RaL < 7 · 10

Horizontal Cavity 1/3 0.074 All properties evaluated at NuL = 0.069Ra P r Heated from Below L average temp. between hot and cold plates 3 1 0.28 −0.25 10 < RaL < 10 0 Heat transfer on P r H H NuL = 0.22 RaL 2 ≤ ≤ 10 Vertical Surfaces 0.2 + P r L L P r ≤ 105

3 RaLP r 0.29 10 < 0.2+P r Heat transfer on P r H NuL = 0.18 RaL 1 ≤ ≤ 2 Vertical Surfaces 0.2 + P r L 10−3 ≤ P r ≤ 105 4 7 −0.3 10 < RaL < 10 Heat transfer on 0.25 0.012 H H NuL = 0.42Ra P r 10 ≤ ≤ 40 Vertical Surfaces L L L 1 ≤ P r ≤ 2 · 104 6 9 10 < RaL ≤ 10 Heat transfer on 1/3 H NuL = 0.046Ra 1 ≤ ≤ 40 Vertical Surfaces L L 1 ≤ P r ≤ 20

5.9 Concentric Cylinders For Cylinders we use an eﬀective thermal conductivity

k P r 1/4 eff = 0.386 Ra1/4 k 0.861 + P r c

The Rayleigh number uses the corrected length

2 [ln(r /r )]4/3 L = o i c −0.6 −0.6 5/3 (ri + ro ) The Heat Transfer is found as 2πLk (T − T ) q = eff i o ln(ro/ri)

5.10 Concentric Spheres For Spheres we use an eﬀective thermal conductivity

k P r 1/4 eff = 0.74 Ra1/4 k 0.861 + P r s

The Rayleigh number uses the corrected length

4/3 1 − 1 ri ro Ls = 1/3 −7/5 −7/5 5/3 2 (ri + ro ) The Heat Transfer is found as 4πLk (T − T ) q = eff i o (1/ri) − (1/ro)

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