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

1

Dimensionless Parameters

Table 1: Dimensionless Parameters

α k ρcp Thermal diffusivity Cf τs ρu2 ∞/2

Skin Friction Coefficient

Le α

DAB

Lewis Number - heat transfer vs. mass transport

N u hL kf

Nusselt Number - Dimensionless Heat Transfer

P e P e = RexP r Peclet Number

P r ν α=

µCp

k Prandtl Number - momentum diffusivity vs. thermal diffusivity Re ρu∞x

µ = u∞x

ν Reynolds Number - Inertia vs. Viscosity

Sc ν

DAB

Schmidt Number momentum vs. mass transport

Sh hmL DAB

Sherwood Number - Dimensionless Mass Transfer

St h ρV cp

= N uL ReLP r

Stanton Number - Modified Nusselt Number

Stm

hm

V = ShL

ReLSc

Stanton mass Number - Modified Sherwood Number

2

Boundary Layer Analogies - Require Geometric Similarity

Table 2: Boundary Layer Analogies

Heat and Mass Analogy

N u P rn = Sh Scn hL kP rn = hmL DABScn

Applies always for same geometry, n is positive

Chilton Colburn Heat jH =

Cf

2 = StP r

2/3

0.6 < P r < 60

Chilton Colburn Mass jM =

Cf

2 = StmSc

2/3

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 pu∞/vx

Re < 5E5

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

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

Local Heat Transfer N ux=

hxx k = 0.332Re 0.5 x P r1/3 Re < 5E5 P r ≥ 0.6

Local Mass Transfer Shx=

hm,xx DAB = 0.332Re0.5 x Sc1/3 Re < 5E5 Sc ≥ 0.6 Average Skin Friction Coefficient Cf,x= 1.328Re−0.5x Re < 1

Average Heat Transfer N ux=

hxx k = 0.664Re 0.5 x P r1/3 Isothermal Re < 5E5 P r ≥ 0.6

Average Mass Transfer Shx=

hm,xx DAB = 0.664Re0.5x Sc1/3 Re < 5E5 Sc ≥ 0.6 N ux N ux= 0.565P e0.5x Liquid Metals N ux= 2N ux P r ≤ 0.05 P ex≥ 100 N ux N ux= 0.3387Re0.5_x P r1/3 1 + (0.0468/P r)2/31/4

All Prandtl Numbers P ex≥ 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−0.2_x 5E5 < Re < 108

Heat Transfer N ux= StRexP r = 0.0296Re0.8x P r1/3

5E5 < Re < 108

0.6 < P r < 60

Mass Transfer Shx= StRexSc = 0.0296Re0.8x Sc1/3

5E5 < Re < 108 0.6 < P r < 3000

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 point

has been found, we define A = 0.037Re0.8

x,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 N uL = (0.037Re0.8L − A)P r

1/3 0.6 < P r < 60

5 · 105_{< Re} L< 108

Average Skin Friction Coefficient CfL = 0.074Re

−0.2₋ 2A

ReL

5 · 105_{< Re} L< 108

Average Mass Transfer ShL= (0.037Re0.8L − A)Sc

1/3 0.6 < Sc < 60

5 · 105_{< Re} L< 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 N ux=

N ux|ζ=0

[1 − (ζ/x)0.75_]1/3

laminar

0 < ReL< 5 · 105

Local Heat Transfer N ux=

N ux|ζ=0

1 − (ζ/x)9/101/9

turbulent 5 · 105_{< Re}

L< 108

Average Heat Transfer N uL= N uL|ζ=0_L−ζL

h

1 − (ζ/L)p+1p+2

ip/(p+1) p = 2 Laminar Flow p = 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 =R

0L(Ts− T∞)dx

Table 7: Constant Heat Flux

Local Heat Transfer Laminar N ux= 0.453Re0.5x P r1/3

0 < ReL < 5 · 105

P r > 0.6

Local Heat Transfer Turbulent N ux= 0.0308Re0.8x P r

1/3 ReL> 5 · 105

0.6 < P r < 60

3.5

Cylinder in Cross Flow

For the cylinder in cross flow, we use ReD = ρV D_µ = V D_ν These typically use properties at the film temperature

Tf =

Ts+ T∞

Table 8: Cylinder in Cross Flow

N uD= CRemDP r 1/3

0.7 < P r < 60 C, m are found as functions

of ReD on P426 N uD= CRemDP r n P r P rs 0.25 0.7 < P r < 500 1 < ReD< 106

All properties evaluated at T∞except P rs Uses table 7.4 P428 N uD= 0.3 + 0.62Re0.5 D P r1/3 1 + (0.4/P r)2/31/4 " 1 +  _Re d 282, 000 5/8#4/5 P r > 0.2

3.6

Flow over Spheres

Table 9: Flow over Spheres

N uD= 2 + (0.4Re0.5D + 0.06Re 2/3 D )P r 0_.4 µ µs 1/4 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∞

N uD= 2 + 0.6Re0.5D P r1/3 For Freely Falling Drops

N uD= 2

Infinite Stationary Medium Red→ 0

3.7

Flow Through Banks of Tubes

3.7.1 Geometric Properties

Table 10: Tube Bank Properties

ReD= ρVmaxD µ Vmax= ST ST − D Vi Aligned OR Staggered and SD> ST + D 2 Vmax= ST 2(SD− D) Vi Staggered and SD< ST + D 2

3.7.2 Flow Correlations

Table 11: Flow through banks of tubes

N uD= 1.13C1RemD,maxP r1/3

More than 10 rows of tubes 2000 < ReD,max< 40, 000

P r > 0.7 Coefficients come from

table 7.5 on P438

N uD|(NL<10)= C2N uD|(NL≥10)

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

N uD= CRemD,maxP r0.36

 P r P rs

0.25

C, m comes from Table 7.7 on P440 1000 < ReD,max< 2 · 106

0.7 < P r < 500 More than 20 rows

N uD|(NL<20)= C2N uD|(NL≥20)

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

Log Mean Temp. ∆Tlm=

(Ts− Ti) − (Ts− T o)

lnTs−Ti

Ts−To



Dimensionless Temp Correlation Ts− To Ts− Ti = exp  − πDN ¯h ρV NTSTcP 

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

3.8

Impinging Jets

Heat and mass transfer is measured against the fluid properties at the nozzle exit q00= h(Ts− Te) The Reynolds

and Nusselt numbers are measured using the hydraulic diameter of the nozzle Dh = Ac,e

P The Reynolds number

uses the nozzle exit velocity. All correlations use the target cell region Arwhich 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 N u = P r 0.42_{G A} r,H_D
2Re0.5(1 + 0.005Re0.55)0.5  2000 < Re < 4 · 105 2 < H/D < 12 0.004 < Ar< 0.04 G factor G = 2A0.5_r 1 − 2.2A 0.5 r 1 + 0.2(H/d − 6)Ar0.5 Always Round Nozzle Array N u = P r 0.42_{0.5K A} r,H_D
G Ar,H_D
Re2/3 2000 < Re < 105 2 < H/D < 12 0.004 < Ar< 0.04 K factor K =  1 +_0.6/ArH/D1/2 6−0.05 Always Single Slot Nozzle N u = P r 0.42 3.06 0.5/Ar+ H/W + 2.78 Rem 3000 < Re < 9 · 104 2 < H/D < 10 0.025 < Ar< 0.125 m factor m = 0.695 − "_ 1 4Ar  +  _H 2W 1.33 + 3.06 #−1 Always Slot Nozzle Array N u = P r 0.422 3A 3/4 r,o  _2Re Ar/Ar,o+ Ar,o/Ar 2/3 SH W L ≥ 1 1500 < Re < 4 · 104 2 < H/D < 80 0.008 < Ar< 2.5Ar,o Ar,o Ar,o= h 60 + 4 _2WH − 2
2i−0.5 Always

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− To Ts− Ti = exp  − ¯hAp,t ρViAc,bcp  For Spheres : ¯jH= ¯jm= 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 - 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

Mean Velocity um= ˙ m ρAc ReD ReD≡ ρumD µ = µmD ν turbulent onset @ ReD≈ 2300

Hydrodynamic Entry Length

x_{f d,h} D  lam ≈ 0.05ReD 10 ≤xf d,h D  turb ≤ 60 Velocity Profile u(r) um = 2 " 1 − r r0 2#

Moody Friction Factor

f ≡ −(dp/dx)D ρu2 m/s f = 64 ReD f = 0.316Re−1/4_D Smooth ReD≤ 2 × 104 f = 0.184Re−1/4_D Smooth ReD≥ 2 × 104 f = (0.790ln(ReD) − 1.64)−2 Smooth 3000 ≤ ReD≤ 5 × 106

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

Table 15: Constant Surface Heat Flux

Convective Heat Transfer qconv= q00s(P L) qs00= constant

Mean Temperature Tm(x) = Tm,i+

q_s00P ˙ mcp

x q_s00= 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  −P xh ˙ mcp  Ts= constant

Table 17: Constant External Environment Temperature

Heat Transfer q = U As∆Tlm T∞= constant

Log Mean Temperature ∆To ∆Ti = T∞− Tm(x) T∞− Tm,i = exp  −U As ˙ mcp  T∞= constant 4.1.2 Flow Correlations

Table 18: Fully Developed Flow In Circular Tubes

N uD≡ hD k = 4.36 lamniar fully developed q_s00= constant N uD≡ hD k = 3.66 lamniar fully developed Ts= constant

Table 19: Laminar Entry Region Flow In Circular Tubes N uD≡ hD k = 3.66 + 0.0668(D/L)ReDP r 1 + 0.04[(D/L)ReDP r]2/3 lamniar Ts= constant

(thermal entry length) OR (combined with Pr ≥ 5) N uD≡ hD k = 1.86  ReDP r L/D 1/3_{ µ} µs 0.14 lamniar Ts= constant 0.60 ≤ P r ≤ 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

N uD≡ hD k = 0.023Re 4/5 D P r n Ts> Tm: n = 0.4 Ts< Tm: n = 0.3 turbulent fully developed small temperature diff 0.6 ≤ P r ≤ 160 ReD≥ 10, 000 N uD≡ hD k = 0.027Re 4/5 D P r 1/3 µ µs 0.14 laminar 0.7 ≤ P r ≤ 16, 700 ReD≥ 10, 000 L D ≥ 10 N uD≡ hD k = (f /8)(ReD− 1000)P r 1 + 12.7(f /8)1/2_{(P r}2/3_{− 1)} lamniar 0.5 ≤ P r ≤ 2000 3000 ≤ ReD≤ 5 × 106

Above appropriate for both constant Tsand constant qs00

N uD≡ hD k = 4.82 + 0.0185P e 0.827 D lamniar

NOT liquid metals (3 × 10−3≤ P r ≤ 5 × 10−2₎

q00_s = constant 3.6 × 103_{≤ Re} D≤ 9.05 × 105 102_{≤ P e} D≤ 104 N uD≡ hD k = 5.0 + 0.025P e 0.8 D

similarly as immediately above Ts= constant

100 ≤ P eD

4.2

Non-Circular Tubes

4.2.1 Properties

Table 21: Flow in Non-Circular Tubes

Hydrodynamic Diameter Dh≡ 4Ac P 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

4.3

Concentric Tube Annulus

4.3.1 Properties

Table 22: Concentric Tube Annulus Properties

Interior heat transfer q_i00= hi(Ts,i− Tm)

Exterior heat transfer q00_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

lamniar fully developed one surface insulated one surface const Ts

N ui= N uii 1 − (q00 o/qi00)θi∗ , N uo= N uoo 1 − (q00_i/q00 o)θo∗

See Table 8.3 for above parameters as a function of Di

Do

laminar q_i00= constant q00

o = constant

4.4

Heat Transfer Enhancement - Tube Coiling

Table 24: Properties for Helically Coiled Tubes

Critical Reynolds Number ReD,c,h= ReD,c[1 + 12(D/C)0.5] ReD,c= 2300 D,C are defined in Figure 8.13 on Page 522 f f = 64 ReD ReD(D/C)1/2≤ 30 f f = 27 Re0 D.725 (D/C)0.1375 30 ≤ ReD(D/C)1/2≤ 300 f f = 7.2 Re0 D.5 (D/C)0.25 300 ≤ ReD(D/C)1/2

Table 25: Correlations for Helically Coiled Tubes

N uD= "_ 3.66 +4.343 a 3 + 1.158 ReD(D/C) 1/2 b 3/2#1/3_{ µ} µs 0.14 a =  1 + 927(C/D) Re2 DP r  0.477 0.005 ≤ P r ≤ 1600 1 ≤ ReDD_C 1/2 ≤ 1000

4.5

Internal Convection Mass Transfer

Table 26: Properties for Internal Convection Mass Transfer

Mean

Species Density ρA,m= R

Ac(ρAu)dAc

umAc

Any Shape

Mean

Species Density ρA,m= 2 umro2

Rro

0 (ρAur)dr Circular Tube

Local Mass Flux n 00 A= hm(ρA,s− ρA,m) Total Mass Flux nA= hmAs∆ρA,lm nA= ˙ m ρ(ρA,o− ρA, i) Log Mean Concentration 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 ˙ m x  Sherwood Number ShD= hmD DAB ShD= hmD DAB

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

5.1

Natural Convection, Vertical Plate

Table 27: Natural Convection, Vertical Plate

Laminar Heat Transfer N ux=

 Grx 4 1/4 g(P r) uses g below g factor g(P r) = 0.75P r 0.5 (0.609 + 1.221P r0.5_{+ 1.238P r)}1/4 0 < P r < ∞ Average Laminar N uL= 4 3  Grx 4 1/4 g(P r) laminar

Better avg. Heat Transfer N uL =

" 0.825 + 0.387Ra 1/6 l 1 + (0.492/P r)9/168/27 #2

Applies for all RaL

Better avg. Laminar Heat Transfer N uL= 0.68 +

0.670Ra1/4_l

1 + (0.492/P r)9/164/9 RaL< 10

9

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

These correlations use L =As

P

Table 28: Natural Convection, Horizontal Plate

Upper Surface Hot Plate

Lower Surface Cold Plate N uL = 0.54Ra

1/4

L 10

4_{< Ra} L< 107

Upper Surface Hot Plate

Lower Surface Cold Plate N uL = 0.15Ra

1/3

L 10

7_{< Ra}

L< 1011

Lower Surface Hot Plate

Upper Surface Cold Plate N uL = 0.27Ra

1/4

L 105< RaL< 1010

5.4

Long Horizontal Cylinder

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

N uD= " 0.60 + 0.387Ra 1/6 D 1 + (0.559/P r)9/168/27 #2

5.5

Spheres

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

Symmetrically Heated Isothermal Plates N us= 1 24Ras  S L   1 − exp  − 35 Ras(S/L) 0.75 10−1 <S_LRas< 105 Symmetrically Heated Isothermal Plates N us= RAs(S/L) 24 10−1< S_LRas< 105 S L → 0 1 Insulated Plate 2 Isothermal Plate N us= Ras(S/L) 12 10−1< S_LRas< 105 S L → 0 Isothermal / Adiabatic (Better) N us=  _C 1 (RasS/L)2 + C2 (RasS/L)1/2 −1/2 RasS_L ≤ 10

The isothermal correlations use N us=

 _q/A Ts− T∞  S k and Ras= gβ(Ts− T∞)S3 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

Isoflux Plates N us,L,f d= 0.144 [Ra

∗ s(S/L)] 0.5 Uses Ra∗ 1 Isoflux Plate 1 Insulated N us,L,f d= 0.204 [Ra ∗ s(S/L)] 0.5 Uses Ra∗ Isoflux / Adiabatic (Better) N us,L=  _C 1 Ra∗ sS/L + C2 (Ra∗ sS/L)2/5 −1/2 RasS_L ≥ 100

The isoflux corelations use N us,f d=

 _q00 s Ts,L− T∞  S k and Ra ∗ s= gβq00_sS4 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

N us= 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

Table 30: Rectangular Channels

Horizontal Cavity

Heated from Below N uL= 0.069Ra

1/3 L P r

0.074

3 · 105< RaL< 7 · 109

All properties evaluated at average temp. between hot and cold plates

Heat transfer on Vertical Surfaces N uL= 0.22  _{P r} 0.2 + P rRaL 0.28_{ H} L −0.25 10 3_{< Ra} L < 1010 2 ≤ H_L ≤ 10 P r ≤ 105 Heat transfer on Vertical Surfaces N uL= 0.18  _{P r} 0.2 + P rRaL 0.29 10 3_< RaLP r 0.2+P r 1 ≤ H_L ≤ 2 10−3 ≤ P r ≤ 105 Heat transfer on

Vertical Surfaces N uL= 0.42Ra

0.25 L P r 0.012 H L −0.3 10 4_{< Ra} L < 107 10 ≤ H_L ≤ 40 1 ≤ P r ≤ 2 · 104 Heat transfer on

Vertical Surfaces N uL= 0.046Ra

1/3 L 106_{< Ra} L ≤ 109 1 ≤ H L ≤ 40 1 ≤ P r ≤ 20

5.9

Concentric Cylinders

For Cylinders we use an effective thermal conductivity

kef f k = 0.386  _{P r} 0.861 + P r 1/4 Ra1/4c

The Rayleigh number uses the corrected length

Lc=

2 [ln(ro/ri)] 4/3

(r−0.6_i + r−0.6o )5/3

The Heat Transfer is found as

q = 2πLkef f(Ti− To) ln(ro/ri)

5.10

Concentric Spheres

For Spheres we use an effective thermal conductivity

kef f k = 0.74  _{P r} 0.861 + P r 1/4 Ra1/4_s

The Rayleigh number uses the corrected length

Ls=  1 ri − 1 ro 4/3 21/3_(r−7/5 i + r −7/5 o )5/3

The Heat Transfer is found as

q = 4πLkef f(Ti− To) (1/ri) − (1/ro)