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 JRB, ASR MEAM333 - Convection Correlations 1 Dimensionless Parameters Table 1: Dimensionless Parameters k α Thermal diffusivity ρcp τs Cf 2 Skin Friction Coefficient ρu1=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 1 = 1 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 2/17 JRB, ASR MEAM333 - Convection Correlations 3 External Flow T + T These typically use properties at the film temperature T = s 1 f 2 3.1 External Flow for a Flat Plate T + T These use properties at the film temperature T = s 1 f 2 Table 3: Flat Plate Isothermal Laminar Flow 5:0 Flat plate Boundary Layer Thickness δ = p Re < 5E5 u1=vx p Local Shear Stress τs = 0:332u1 ρµu1=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 3/17 JRB, ASR MEAM333 - Convection Correlations 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 1 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 1 s 1 f 2 Table 6: Unheated Starting Length Nu j laminar Local Heat Transfer Nu = x ζ=0 x 1=3 5 [1 − (ζ=x)0:75] 0 < ReL < 5 · 10 Nu j 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 = NuLjζ=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 − T1)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 1 f 2 4/17 JRB, ASR MEAM333 - Convection Correlations 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 T1 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 T1 0:5 1=3 NuD = 2 + 0:6ReD P r For Freely Falling Drops Infinite Stationary Medium NuD = 2 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 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 6/17 JRB, ASR MEAM333 - Convection Correlations 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 NuDj(NL<10) = C2NuDj(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 NuDj(NL<20) = C2NuDj(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 7/17 JRB, ASR MEAM333 - Convection Correlations 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.