Heat and Mass Correlations

Heat and Mass Correlations Alexander Rattner, Jonathan Bohren November 13, 008 Contents 1 Dimensionless Parameters Boundary ayer Analogies - Require Geometric Similarity 3 External Flow 3 3.1 External Flow for a Flat Plate...................................... 3 3. Mixed Flow Over a plate.......................................... 4 3.3 Unheated Starting ength......................................... 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. 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. Flow Correlations......................................... 10 4. Non-Circular Tubes............................................ 1 4..1 Properties.............................................. 1 4.. Flow Correlations......................................... 1 4.3 Concentric Tube Annulus......................................... 13 4.3.1 Properties.............................................. 13 4.3. 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. Natural Convection, Inclined Plate.................................... 15 5.3 Natural Convection, Horizontal Plate................................... 15 5.4 ong 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

1 Dimensionless Parameters α C f e Nu k ρc p τ s ρu / α D AB h k f Table 1: Dimensionless Parameters Thermal diffusivity Skin Friction Coefficient ewis Number - heat transfer vs. mass transport Nusselt Number - Dimensionless Heat Transfer P e P e = Re x P r Peclet Number ν P r α = µc p Prandtl Number - momentum diffusivity vs. thermal diffusivity k ρu x Re = u x Reynolds Number - Inertia vs. Viscosity µ ν ν Sc Schmidt Number momentum vs. mass transport Sh St St m D AB h m D AB h = Nu ρv c p Re P r h m V = Sh Re Sc Sherwood Number - Dimensionless Mass Transfer Stanton Number - Modified Nusselt Number Stanton mass Number - Modified Sherwood Number Boundary ayer Analogies - Require Geometric Similarity Table : Boundary ayer Analogies Heat and Mass Analogy Nu P r n = Sh Sc n h kp r n = h m D AB Sc n Applies always for same geometry, n is positive Chilton Colburn Heat Chilton Colburn Mass j H = C f = StP r/3 0.6 < P r < 60 j M = C f = St msc /3 0.6 < Sc < 3000 /17

3 External Flow These typically use properties at the film temperature T f = T s + T 3.1 External Flow for a Flat Plate These use properties at the film temperature T f = T s + T Table 3: Flat Plate Isothermal aminar Flow Flat plate Boundary ayer Thickness δ = 5.0 u /vx Re < 5E5 ocal Shear Stress τ s = 0.33u ρµu /x Re < 5E5 ocal Skin Friction Coefficient C f,x = 0.664Re 0.5 x Re < 1 ocal Heat Transfer ocal Mass Transfer Nu x = h xx k = 0.33Re0.5 x P r 1/3 Re < 5E5 P r 0.6 Sh x = h m,xx D AB = 0.33Re 0.5 x Sc 1/3 Re < 5E5 Sc 0.6 Average Skin Friction Coefficient C f,x = 1.38Re 0.5 x Re < 1 Average Heat Transfer Average Mass Transfer Nu x = h xx k = 0.664Re0.5 x P r 1/3 Re < 5E5 Isothermal P r 0.6 Sh x = h m,xx D AB = 0.664Re 0.5 x Sc 1/3 Re < 5E5 Sc 0.6 iquid Metals Nu x Nu x = 0.565P e 0.5 x Nu x Nu x = 0.3387Re 0.5 x P r 1/3 1 + 0.0468/P r) /3 ] 1/4 Nu x = Nu x P r 0.05 P e x 100 All Prandtl Numbers P e x 100 Table 4: Turbulent Flow Over an Isothermal Plate Re x > 5 10 5 Skin Friction Coefficient C f,x = 0.059Rex 0. 5E5 < Re < 10 8 Boundary ayer Thickness δ = 0.37xRex 0. 5E5 < Re < 10 8 Heat Transfer Nu x = StRe x P r = 0.096Re 0.8 x P r 1/3 5E5 < Re < 10 8 0.6 < P r < 60 Mass Transfer Sh x = StRe x Sc = 0.096Re 0.8 x Sc 1/3 5E5 < Re < 10 8 0.6 < P r < 3000 3/17

3. Mixed Flow Over a plate If transition occurs at xc 0.95 The laminar plate model may be used for h. Once the critical transition point has been found, we define A = 0.037Re 0.8 x,c 0.664Rex,c 0.5 These typically use properties at the film temperature T f = T s + T Table 5: Mixed Flow Over an Isothermal Plate Average Heat Transfer Nu = 0.037Re 0.8 A)P r1/3 0.6 < P r < 60 5 10 5 < Re < 10 8 Average Skin Friction Coefficient C f = 0.074Re 0. A Re 5 10 5 < Re < 10 8 Average Mass Transfer Sh = 0.037Re 0.8 A)Sc1/3 0.6 < Sc < 60 5 10 5 < Re < 10 8 3.3 Unheated Starting ength Here the plate has T s = T until x = ζ These typically use properties at the film temperature T f = T s + T Table 6: Unheated Starting ength ocal Heat Transfer Nu x ζ=0 laminar Nu x = 1 ζ/x) 0.75 ] 1/3 0 < Re < 5 10 5 ocal Heat Transfer Nu x ζ=0 Nu x = ] turbulent 1 ζ/x) 9/10 1/9 5 10 5 < Re < 10 8 Average Heat Transfer Nu = Nu ζ=0 ζ ] 1 ζ/) p+1 p/p+1) p+ p = aminar 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 = 0 T s T )dx Table 7: Constant Heat Flux ocal Heat Transfer aminar Nu x = 0.453Re 0.5 x P r 1/3 0 < Re < 5 10 5 P r > 0.6 ocal Heat Transfer Turbulent Nu x = 0.0308Re 0.8 x P r 1/3 Re > 5 10 5 0.6 < P r < 60 3.5 Cylinder in Cross Flow For the cylinder in cross flow, we use Re D = ρv D µ = V D ν T f = T s + T These typically use properties at the film temperature 4/17

Table 8: Cylinder in Cross Flow Nu D = CRe m D P r1/3 0.7 < P r < 60 C, m are found as functions of Re D on P46 0.7 < P r < 500 ) 0.5 1 < Re D < 10 6 P r Nu D = CRe m D P rn All properties evaluated at P r s T except P r s Uses table 7.4 P48 Nu D = 0.3 + 0.6Re0.5 D P ) ] 5/8 4/5 r1/3 Red ] 1 + P r > 0. 1 + 0.4/P r) /3 1/4 8, 000 3.6 Flow over Spheres Nu D = + 0.4Re 0.5 D Nu D = + 0.6Re 0.5 D P r1/3 Nu D = Table 9: Flow over Spheres 0.71 < P r < 380 ) 1/4 3.5 < P r < 6.6 10 4 µ + 0.06Re/3 D )P r0.4 1.0 < µ/µ µ s ) < 3. s All properties except µ s are evaluated at T For Freely Falling Drops Infinite Stationary Medium Re d 0 5/17

3.7 Flow Through Banks of Tubes 3.7.1 Geometric Properties Re D = ρv maxd µ V max = V max = Table 10: Tube Bank Properties S T S T D V i Aligned OR Staggered and S D > S T + D S T S D D) V i Staggered and S D < S T + D Figure 1: Tube bank geometries for aligned a) and staggered b) banks 6/17

3.7. Flow Correlations Nu D = 1.13C 1 Re m D,max P r1/3 Nu D N <10) = C Nu D N 10) Nu D = CRe m D,max P r0.36 P r P r s Nu D N <0) = C Nu D N 0) Table 11: Flow through banks of tubes More than 10 rows of tubes 000 < Re D,max < 40, 000 P r > 0.7 Coefficients come from table 7.5 on P438 C comes from Table 7.6 on P439 000 < Re D,max < 40, 000 P r > 0.7 Coefficients come from table 7.5 on P438 ) 0.5 C, m comes from Table 7.7 on P440 1000 < Re D,max < 10 6 0.7 < P r < 500 More than 0 rows For the above correlation C comes from Table 7.8 on P440 000 < Re D,max < 40, 000 P r > 0.7 Table 1: Flow through banks of tubes og Mean Temp. T lm = T s T i ) T s T o) ) T ln s T i T s T o T s T o Dimensionless Temp Correlation = exp πdn h ) T s T i ρv N T S T c P N - total number of tubes, N T - total number of tubes in transverse plane Heating Per Unit ength q = N hπd T lm 7/17

3.8 Impinging Jets Heat and mass transfer is measured against the fluid properties at the nozzle exit q = ht s T e ) The Reynolds and Nusselt numbers are measured using the hydraulic diameter of the nozzle D h = Ac,e P The Reynolds number uses the nozzle exit velocity. All correlations use the target cell region A r 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 Single Round Nozzle Nu = P r 0.4 G A r, H D G factor G = A 0.5 r Round Nozzle Array K factor K = Single Slot Nozzle m factor Slot Nozzle Array Table 13: Impinging Jets ) Re 0.5 1 + 0.005Re 0.55 ) 0.5] 000 < Re < 4 10 5 1.A 0.5 r 1 + 0.H/d 6)Ar 0.5 Always Nu = P r 0.4 0.5K A r, H D ) G Ar, H D ) Re /3 ) ] 6 0.05 1 + H/D 0.6/Ar 1/ Nu = P r 0.4 3.06 0.5/A r + H/W +.78 Rem m = 0.695 1 4A r Nu = P r 0.4 3 A3/4 r,o ) ) 1.33 H + + 3.06 W Re A r /A r,o + A r,o /A r < H/D < 1 0.004 < A r < 0.04 000 < Re < 10 5 < H/D < 1 0.004 < A r < 0.04 Always ] 1 Always 3000 < Re < 9 10 4 < H/D < 10 0.05 < A r < 0.15 ) /3 SH W 1 1500 < Re < 4 10 4 < H/D < 80 A r,o A r,o = 60 + 4 H W ) ] 0.5 0.008 < A r <.5A r,o Always 8/17

3.9 Packed Beds For packed beds, the heat transfer depends on the total particle surface area A p,t q = ha p,t T lm The outlet temperature can be determined from the log mean relation T s T o = exp ha ) p,t T s T i ρv i A c,b c p For Spheres: ɛ j H = ɛ j m =.06Re 0.575 D where Pr or Sc 0.7 and 90 < Re D < 4000 For non spheres multiply the right hand side by a factor - uniform cylinders of = D use 0.71, for uniform cubes use 0.71 ɛ 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 Re D Hydrodynamic Entry ength Velocity Profile Moody Friction Factor u m = ṁ ρa c Re D ρu md µ xfd,h ) 0.05Re D D lam xfd,h ) 10 60 D turb ) ] ur) r = 1 u m = µ md ν turbulent onset @ Re D 300 r 0 f dp/dx)d ρu m/s f = 64 Re D f = 0.316Re 1/4 D f = 0.184Re 1/4 D f = 0.790lnRe D ) 1.64) Power for Pressure Drop P = p) = ṁ ρ Smooth Re D 10 4 Smooth Re D 10 4 Smooth 3000 Re D 5 10 6 9/17

Table 15: Constant Surface Heat Flux Convective Heat Transfer q conv = q s P ) q s = constant Mean Temperature T m x) = T m,i + q s P x ṁc p q s = constant Table 16: Constant Surface Temperature Convective Heat Transfer q conv = ha s T lm T s = constant og Mean Temperature T lm T o T i ln T o / T i ) T o = T s T m x) = exp P xh ) T s = constant T i T s T m,i ṁc p Table 17: Constant External Environment Temperature Heat Transfer q = UA s T lm T = constant T og Mean Temperature o = T T m x) = exp UA ) s T T i T T m,i ṁc = constant p 4.1. Flow Correlations Table 18: Fully Developed Flow In Circular Tubes Nu D hd k = 4.36 lamniar fully developed q s = constant Nu D hd k = 3.66 lamniar fully developed T s = constant 10/17

Table 19: aminar Entry Region Flow In Circular Tubes Nu D hd k = 3.66 + 0.0668D/)Re D P r 1 + 0.04D/)Re D P r] /3 Nu D hd k lamniar T s = constant thermal entry length) OR combined with Pr 5) lamniar ) 1/3 ) T s = constant 0.14 = 1.86 ReD P r µ 0.60 P r 5 ) /D µ s µ 0.0044 µ s 9.75 All properties evaluated at the mean temperature T m = T m,i + T m,o )/ Table 0: Turbulent Flow In Circular Tubes Nu D hd k Nu D hd k = 0.03Re4/5 D P rn T s > T m : n = 0.4 T s < T m : n = 0.3 ) 0.14 µ = 0.07Re4/5 D P r1/3 µ s turbulent fully developed small temperature diff 0.6 P r 160 Re D 10, 000 laminar 0.7 P r 16, 700 Re D 10, 000 D 10 Nu D hd k = f/8)re D 1000)P r 1 + 1.7f/8) 1/ P r /3 1) Nu D hd k lamniar 0.5 P r 000 3000 Re D 5 10 6 Above appropriate for both constant T s and constant q s = 4.8 + 0.0185P e0.87 D lamniar NOT liquid metals 3 10 3 P r 5 10 ) q s = constant 3.6 10 3 Re D 9.05 10 5 10 P e D 10 4 Nu D hd similarly as immediately above = 5.0 + 0.05P e0.8 D T k s = constant 100 P e D All properties evaluated at the mean temperature T m = T m,i + T m,o )/ 11/17

4. Non-Circular Tubes 4..1 Properties Table 1: Flow in Non-Circular Tubes Hydrodynamic Diameter D h 4A c P Re Re Dh ρu md h = µ md h Dh µ ν turbulent onset @ Re Dh 300 All properties evaluated at the mean temperature T m = T m,i + T m,o )/ 4.. Flow Correlations Figure : Nusselt numbers and friction factors for fully developed laminar flow in tubes of differing cross-section 1/17

4.3 Concentric Tube Annulus 4.3.1 Properties Table : Concentric Tube Annulus Properties Interior heat transfer q i it s,i T m ) Exterior heat transfer q o = h o T s,o T m ) Hydrodynamic Diameter D h = D o D i 4.3. Flow Correlations Table 3: Correlations for Concentric Tube Annulus Nu i = See Table 8. on Page 50 Nu ii 1 q o /q i, Nu o = )θ i Nu oo 1 q i /q o )θ o See Table 8.3 for above parameters as a function of Di D o lamniar fully developed one surface insulated one surface const T s laminar = constant q i q o = constant 4.4 Heat Transfer Enhancement - Tube Coiling Critical Reynolds Number Table 4: Properties for Helically Coiled Tubes Re D,c,h = Re D,c 1 + 1D/C) 0.5 ] Re D,c = 300 D,C are defined in Figure 8.13 on Page 5 f f = 64 Re Re D D/C) 1/ 30 D 7 f f = Re 0 30 Re D D/C) 1/ 300 D.75D/C)0.1375 f f = 7. Re 0 D.5D/C)0.5 300 Re D D/C) 1/ Table 5: Correlations for Helically Coiled Tubes Nu D = 3.66 + 4.343 ) 3 ReD D/C) 1/ ) 3/ ] 1/3 µ + 1.158 a b a = 1 + 97C/D) ) Re D P r b = 1 + 0.477 P r µ s ) 0.14 0.005 P r 1600 D 1/ 1 Re D C 1000 13/17

4.5 Internal Convection Mass Transfer Mean Species Density Mean Species Density ocal Mass Flux Table 6: Properties for Internal Convection Mass Transfer A ρ A,m = c ρ A u)da c Any Shape u m A c ρ A,m = ro u m ro 0 ρ Aur)dr Circular Tube n A = h mρ A,s ρ A,m ) Total Mass Flux og Mean Concentration Difference Sherwood Number ρ A,lm = ρ A x) ρ A,i n A = h m A s ρ A,lm n A = ṁ ρ ρ A,o ρa, i) ρ A,o ρ A,i ln ρ A,o / ρ A,i ) = ρ A,s ρ A,m x) ρ A,s ρ A,m,i Sh D = h md D A B Sh D = h md D A B = exp h ) mρp ṁ x The concentration entry length x fd,c can be determined with the mass transfer analogy and the same function used to determine x fd,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. happens around Ra 10 9 Transition to turbulent flow 14/17

5.1 Natural Convection, Vertical Plate Table 7: Natural Convection, Vertical Plate ) 1/4 Grx aminar Heat Transfer Nu x = gp r) uses g below 4 g factor gp r) = 0.75P r 0.5 0.609 + 1.1P r 0.5 + 1.38P r) 1/4 0 < P r < Average aminar Nu = 4 ) 1/4 Grx gp r) laminar 3 4 ] 0.387Ra 1/6 l Better avg. Heat Transfer Nu = 0.85 + ] Applies for all Ra 1 + 0.49/P r) 9/16 8/7 Better avg. aminar Heat Transfer Nu = 0.68 + 0.670Ra 1/4 l 1 + 0.49/P r) 9/16 ] 4/9 Ra < 10 9 5. 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 cosθ) substituted into Ra 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 = As P Table 8: Natural Convection, Horizontal Plate Upper Surface Hot Plate ower Surface Cold Plate Upper Surface Hot Plate ower Surface Cold Plate ower Surface Hot Plate Upper Surface Cold Plate Nu = 0.54Ra 1/4 10 4 < Ra < 10 7 Nu = 0.15Ra 1/3 10 7 < Ra < 10 1 1 Nu = 0.7Ra 1/4 10 5 < Ra < 10 1 0 5.4 ong Horizontal Cylinder Assumes isothermal cylinder. The following correlation applies for Ra D < 10 1 Nu D = 0.60 + ] 0.387Ra 1/6 D ] 1 + 0.559/P r) 9/16 8/7 5.5 Spheres For P r > 0.7 and Ra D < 10 1 1 Nu D = + 0.589Ra 1/4 D 1 + 0.469/P r) 9/16 ] 4/9 15/17

5.6 Vertical Channels This section describes correlations for natural convection between to parralel plates. It uses Ra s which uses the plate separation for the length scale. I believe that the convection area is the surface area where heating/cooling happens. Symmetrically Heated Isothermal Plates Symmetrically Heated Isothermal Plates 1 Insulated Plate Isothermal Plate Isothermal / Adiabatic Better) Nu s = 1 4 Ra s Table 9: Vertical Channels ) S 1 exp Nu s = RA ss/) 4 Nu s = Ra ss/) 1 35 Ra s S/) C 1 Nu s = Ra s S/) + C Ra s S/) 1/ ) q/a S The isothermal correlations use Nu s = T s T Symmetric Isoflux Plates 1 Isoflux Plate 1 Insulated Isoflux / Adiabatic Better) )] 0.75 10 1 < S Ra s < 10 5 10 1 < S Ra s < 10 5 S 0 10 1 < S Ra s < 10 5 S 0 ] 1/ Ra s S 10 k and Ra s = gβt s T )S 3 αν The better isothermal correlation uses C 1 = 576, C =.87 for Symmetric isothermal Plates C 1 = 144, C =.87 for isothermal and adiabatic Plates Nu s,,fd = 0.144 Ra ss/)] 0.5 Nu s,,fd = 0.04 Ra ss/)] 0.5 Nu s, = C 1 Ra ss/ + C Ra ss/) /5 Uses Ra Uses Ra ] 1/ Ra s S 100 q ) s S The isoflux corelations use Nu s,fd = T s, T k and Ra s = gβq kαν The better isoflux correlation uses C 1 = 48, C =.51 for Symmetric isoflux Plates C 1 = 4, C =.51 for isoflux and adiabatic Plates s S 4 5.7 Inclined Channels For plates inclined less than 45 degrees from the vertical Nu s = 0.645 Ra s S/)] 1/4 Fluid properties are evaluated at T = Ts+T This requires Ra s S/) > 00 5.8 Rectangular Cavities For a channel with flow through the Hx plane, no advection happens unless Ra > 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

Table 30: Rectangular Channels Horizontal Cavity Heated from Below Heat transfer on Vertical Surfaces Heat transfer on Vertical Surfaces Heat transfer on Vertical Surfaces Heat transfer on Vertical Surfaces Nu = 0.069Ra 1/3 P r0.074 P r Nu = 0. 0. + P r Ra ) 0.8 H P r Nu = 0.18 0. + P r Ra 3 10 5 < Ra < 7 10 9 All properties evaluated at average temp. between hot and cold plates ) 0.5 10 3 < Ra < 10 1 0 H 10 P r 10 5 ) 0.9 10 3 < Ra P r 0.+P r 1 H 10 3 P r 10 5 ) 0.3 10 4 < Ra < 10 7 H Nu = 0.4Ra 0.5 P r0.01 10 H 40 1 P r 10 4 Nu = 0.046Ra 1/3 10 6 < Ra 10 9 1 H 40 1 P r 0 5.9 Concentric Cylinders For Cylinders we use an effective thermal conductivity k eff k The Rayleigh number uses the corrected length = 0.386 P r 0.861 + P r ) 1/4 Ra 1/4 c The Heat Transfer is found as c = lnr o/r i )] 4/3 r 0.6 i + r 0.6 o ) 5/3 q = πk eff T i T o ) lnr o /r i ) 5.10 Concentric Spheres For Spheres we use an effective thermal conductivity k eff k The Rayleigh number uses the corrected length The Heat Transfer is found as = 0.74 s = P r 0.861 + P r ) 4/3 1 r i 1 r o ) 1/4 Ra 1/4 s 1/3 r 7/5 i + r 7/5 o ) 5/3 q = 4πk eff T i T o ) 1/r i ) 1/r o ) 17/17