External Flow Correlations (Average, Isothermal Surface)

Flat Plate Correlations Flow Average Nusselt Number Restrictions Conditions Note: All fluid properties are Laminar 1/ 2 1/3 Pr  0.6 evaluated at film NuL  0.664ReL Pr temperature for flat plate correlations. 4/5 1/3 NuL  0.037 ReL  A Pr 0.6  Pr  60 Turbulent 4/5 1/ 2 8 where A  0.037 Re x,c 0.664Re x,c Re x,c  ReL 10

Cylinders in Cross Flow Cylinder Reynolds Cross Number Average Nusselt Number Restrictions Note: All fluid properties Section Range are evaluated at film temperature for cylinder V 0.4-4 0.330 1/3 in cross flow correlations. D NuD  0.989ReD Pr Pr  0.7

V D 4-40 Nu  0.911Re0.385 Pr1/3 Pr  0.7 D D Alternative Correlations for Circular Cylinders in Cross Flow: V 0.466 1/3

D 40-4,000 NuD  0.683ReD Pr . The Zukauskas correlation (7.53) and V 4,000- 0.618 1/3 the Churchill and Bernstein correlation D Nu  0.193Re Pr 40,000 D D (7.54) may also be used

V 40,000- 0.805 1/3 D Nu  0.027 Re Pr 400,000 D D

V 6,000- D 0.59 1/3 NuL  0.304ReD Pr gas flow 60,000 Freely Falling Liquid Drops

V 5,000- 0.66 1/3 D Nu  0.158Re Pr gas flow Average Nusselt Number 60,000 D D

V 5,200- 0.638 1/3 1/ 2 1/3 D Nu  0.164Re Pr gas flow NuD  2  0.6ReD Pr 20,400 D D Note: All fluid properties are evaluated V 20,400- 0.78 1/3 D Nu  0.039Re Pr gas flow at T for the falling drop correlation. 105,000 D D ∞

V 4,500- 0.638 1/3 D gas flow 90,700 NuD  0.150ReD Pr

Flow Around a Sphere

Average Nusselt Number Restrictions Note: For flow around a sphere, all fluid properties, 0.71 Pr  380 except μ , are evaluated at T . 1/ 4 s ∞ μ is evaluated at T .    4 s S 1/ 2 2/3 0.4   3.5  Re  7.610 NuD  2  0.4ReD  0.06ReD Pr   D  s  1.0   / s  3.2 Internal Flow Correlations (Local, Fully Developed Flow)

Note: For all local correlations, fluid properties are evaluated at Tm. For average correlations, fluid properties are evaluated at the average of inlet and outlet Tm. If the tube is much longer than the thermal entry length, average correlation ≈ local correlation.

Laminar Flow in Circular and Noncircular Tubes hD Nu  h D k b Uniform Heat Flux Cross Section Uniform Surface f Re a Temperature Dh

Turbulent Flow in Circular Tubes

Local Nusselt Number Restrictions

4/5 n 0.6  Pr 160 NuD  0.023ReD Pr Liquid Metals, Turbulent Flow, Constant Ts n  0.40 for T  T ReD 10,000 s m n  0.30 for Ts  Tm L / D10 Local Nusselt Number Restrictions

Turbulent Flow in Noncircular Tubes 0.8 Pe 100 NuD  5.0 0.025PeD D

For turbulent flow in noncircular tubes, D in the PeD  Re D Pr

table above may be replaced by Dh=4Ac / P Note: Only use the correlation in the box directly above for liquid metals. The other correlations on Alternative Correlations for Turbulent this page are not applicable to liquid metals. Flow in Circular Tubes:

• The Sieder and Tate Correlation (8.61) is recommended for flows with large property variations

• Another alternate correlation that is more complex but more accurate is provided by Gnielinski (8.62).

Combined Internal/External Flow Correlations (Average)

Tube banks and packed beds have characteristics of both internal and external flow. The flow is internal in that the fluid flows inside the tube bank/packed bed, exhibits exponential temperature profiles of the mean temperature, and has heat transfer governed by a log mean temperature difference. The flow is external in that it flows over tubes/packed bed particles and that the characteristic dimension in the Reynolds number is based on tube/particle diameter.

Tube Bank Correlation

Average Nusselt Number Restrictions Note: For tube banks with fewer than 20 rows, multiply the average Nusselt number from the table 1/ 4 N  20 L at left by the correction factor C2 in Table 7.6. m 0.36 Pr  This correction is valid if Re is > 1,000. Nu  C Re Pr   0.7  Pr  500 D,max D D,max    Prs  6 10  ReD,max  210

Packed Bed Correlation

Average Nusselt Number Restrictions

0.575 Pr(or Sc)  0.7  jH   jM  2.06Re D 90  Re  4,000 D where h 2/3 jH  Pr Vc p

h j  m Sc2/3 M V External Free Convection Correlations (Average, Isothermal)

Evaluate all fluid properties at the film temperature Tf = (T∞ + Ts ) / 2 .

Vertical Plate, Vertical Cylinder, Top Side of Inclined Cold Plate, Bottom Side of Inclined Hot Plate

Average Nusselt Number Restrictions

Vertical plate: no restrictions Alternative Correlation D 35 for Vertical Plate: 2 Vertical cylinder:  1/6 1/ 4   L GrL  0.387Ra L  . Equation (9.27) is Nu  0.825  L  8/ 27  slightly more accurate for  9/16  Top Surface of Inclined Cold Plate /  1 0.492 / Pr   Bottom Surface of Inclined Hot Plate: laminar flow.

• Replace g with g cos q in RaL

• Valid for 0 q  60

Horizontal Plate

Orientation Average Nusselt Number Restrictions

1/ 4 4 7 , NuL  0.54Ra L 10  Ra L 10 Pr  0.7 Upper surface of hot plate or A lower surface of cold plate L  s 1/3 7 11 , NuL  0.15Ra L 10  Ra L 10 all Pr P

Lower surface of hot plate or upper surface of cold plate 1/5 4 9 , NuL  0.52Ra L 10  Ra L 10 Pr  0.7

Curved Shapes

Shape Average Nusselt Number Restrictions

Alternative Correlation 2 for Long Horizontal Long  0.387Ra1/6  Horizontal D 12 Cylinder: NuD  0.60  8/ 27  Ra D 10 Cylinder  1 0.559 / Pr 9/16        . The Morgan correlation (9.33) may also be used.

1/ 4 0.589Ra D NuD  2  Sphere 9/16 4/9 11 1 0.469 / Pr  Ra D 10 Internal Free Convection Correlations

Vertical Parallel Plate Channels (Developing and Fully Developed) Temperature to Boundary Getting q and q ” Nusselt Number Rayleigh Number s evaluate fluid Condition from Nu properties in Ra

Average Nu over whole plate isothermal 1/ 2 3  q / A  S (Ts known gT T S Ts T  C1 C2  s  Nu    on one or RaS  S   T  NuS   2  1/ 2  T T k both plates)   s   2 RaS S / L RaS S / L 

Local Nu at x = L isoflux " 1/ 2 " 4  q  S T T   * gq S s s,L  (qs” known C C s Nu    T  1 2 RaS  S,L   on one or NuS,L     T T k 2 Ra * S / L * 2/5 k  s,L   both plates)  S RaS S / L 

S = plate spacing; T∞=inlet temperature (same as ambient); Ts,L=surface temperature at x=L

C1 and C2 are given for four different sets of surface 1) boundary conditions. Use the isothermal equation for conditions 1 and 3; isoflux equation for conditions 2 and 4. 2) T T = T T q” =0 q” q” =0 3) s1 s2 s1 q”s1 q”s2=q”s1 s1 s2 s1 s2 1) 2) 3) 4) 4)

Vertical Rectangular Cavity H Average Nusselt Number Restrictions w 1 H L 2 L 0.29  Pr  103  Pr 105 Alternative Correlation for Vertical NuL  0.18 Ra L   Pr 0.2  Rectangular Cavity: Ra Pr 103  L 0.2  Pr . Eq. (9.53) covers a wide range of aspect ratios but is more restrictive on Ra and Pr 2  H L10 Pr 0.28 H 1/ 4     5 Horizontal Cavity Heated From Below NuL  0.22 Ra L    Pr 10  Pr 0.2   L  3 10 Average Nusselt Number Restrictions 10  Ra L 10 1/3 0.074 Nu  0.069 Ra Pr 5 9 10  H L 40 L L 310  Ra L  710 0.3 1/ 4 0.012 H  4 NuL  0.42 Ra L Pr   1 Pr  210  L  For cavity correlations, evaluate all fluid properties 4 7 10  Ra 10 at the average surface temperature T= (T1 + T2 ) / 2 . L L is the distance between hot and cold walls.

Correlations for Inclined/Tilted Geometries: Correlations for Curved Geometries:

. Inclined parallel plate channels: (9.47) . Space between concentric horizontal cylinders: (9.58) . Tilted rectangular cavities: (9.54)-(9.57) . Space between concentric spheres: (9.61)

Boiling and Condensation

Nucleate Pool Boiling For all condensation correlations below: Evaluate liquid properties at Tf = (Tsat + Ts ) / 2 . 1/ 2 3 Evaluate v and hfg at Tsat .  g(   )  cp,l Te  q"   h l v   s l fg    n  Laminar Film Condensation, Vertical Flat Plate     Cs, f hfg Prl  3 1/ 4 Evaluate liquid and vapor properties at Tsat . l g(l  v )hfg kl  hL  0.943   l (Tsat Ts )L 

hfg  hfg  0.68cp,l (Tsat Ts )

Laminar, Transition, and Film Condensation, Turbulent Film Condensation, Vertical Tube:

Vertical Flat Plate (for l >> v): . Vertical flat plate . Calculate the parameter P using expressions can be (10.42), then solve for hL using the used if d(L) << D/2. appropriate correlation from Evaluate d(L) using (10.43)-(10.45) (10.26).

Laminar Film Condensation, Sphere and Tube

3 1/ 4 Critical Heat Flux l g(l  v )hfg kl  C =0.826 for spheres. hD  C   1/ 4  l (Tsat Ts )D   g(   )   C=0.729 for horizontal q"  Ch  l v max fg v  2  tubes.  v 

Evaluate liquid and vapor properties at Tsat . Laminar Film Condensation, Vertical Tier of N Tubes: C =0.149 for large horizontal plates. C=0.131 for large horizontal cylinders, spheres, . Average heat transfer coefficient of each tube: Eq. (10.49). and many large finite heated surfaces. Inner Surface of Horizontal Tube Film Boiling

1/ 4 Average Nusselt Number Restrictions   3  hconvD g(l  v )hfg D Nu   C  1/ 4 D k  k (T T ) 3 v  v v s sat  l g(l  v )hfg kl  h  0.555   v um,v D  D  (T T )D    35,000 h  h  0.80c (T T )  l sat s     fg fg p,v s sat  v i hfg  hfg  0.375cp,l (Tsat Ts ) Evaluate vapor properties at Tf = (Tsat + Ts ) / 2 . Evaluate l and hfg at Tsat .  v um,v D  Eq. (10.51)    35,000 C =0.67 for spheres. C=0.62 for horizontal cylinders.     v i Radiation should be considered for Ts > 300°C See Eqs. (10.9)-(10.11) Dropwise Condensation

Correlations for Flow Boiling: Average Nusselt Number Restrictions

. External forced convection boiling: (10.12)-(10.14) h  51,104  2044T C 22C  Tsat 100C . Two-phase flow: (10.15)-(10.16) dc sat

hdc  255,510 Tsat 100C