Heat and Mass Correlations

Transcription

1 Heat and Mass Correlations Alexander Rattner, Jonathan Bohren November 13, 008 Contents 1 Dimensionless Parameters Boundary ayer Analogies - Require Geometric Similarity 3 External Flow External Flow for a Flat Plate Mixed Flow Over a plate Unheated Starting ength Plates with Constant Heat Flux Cylinder in Cross Flow Flow over Spheres Flow Through Banks of Tubes Geometric Properties Flow Correlations Impinging Jets Packed Beds Internal Flow Circular Tube Properties Flow Correlations Non-Circular Tubes Properties Flow Correlations Concentric Tube Annulus Properties Flow Correlations Heat Transfer Enhancement - Tube Coiling Internal Convection Mass Transfer Natural Convection Natural Convection, Vertical Plate Natural Convection, Inclined Plate Natural Convection, Horizontal Plate ong Horizontal Cylinder Spheres Vertical Channels Inclined Channels Rectangular Cavities Concentric Cylinders Concentric Spheres

2 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 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 = Re 0.5 x P r 1/ /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 > 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 < < P r < 60 Mass Transfer Sh x = StRe x Sc = 0.096Re 0.8 x Sc 1/3 5E5 < Re < < P r < /17

4 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 < < Re < 10 8 Average Skin Friction Coefficient C f = 0.074Re 0. A Re < Re < 10 8 Average Mass Transfer Sh = 0.037Re 0.8 A)Sc1/3 0.6 < Sc < < Re < 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 < ocal Heat Transfer Nu x ζ=0 Nu x = ] turbulent 1 ζ/x) 9/10 1/ < 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 < P r > 0.6 ocal Heat Transfer Turbulent Nu x = Re 0.8 x P r 1/3 Re > < P r < 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

5 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 P < P r < 500 ) < 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 = Re0.5 D P ) ] 5/8 4/5 r1/3 Red ] 1 + P r > /P r) /3 1/4 8, 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 < µ Re/3 D )P r < 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

6 3.7 Flow Through Banks of Tubes 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

7 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 P < 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 P < Re D,max < < P r < 500 More than 0 rows For the above correlation C comes from Table 7.8 on P < 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

8 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 Re 0.55 ) 0.5] 000 < Re < A 0.5 r H/d 6)Ar 0.5 Always Nu = P r K A r, H D ) G Ar, H D ) Re /3 ) ] H/D 0.6/Ar 1/ Nu = P r /A r + H/W +.78 Rem m = A r Nu = P r A3/4 r,o ) ) 1.33 H W Re A r /A r,o + A r,o /A r < H/D < < A r < < Re < 10 5 < H/D < < A r < 0.04 Always ] 1 Always 3000 < Re < < H/D < < A r < 0.15 ) /3 SH W < Re < < H/D < 80 A r,o A r,o = H W ) ] < A r <.5A r,o Always 8/17

9 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 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 Internal Flow 4.1 Circular Tube 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 ) D turb ) ] ur) r = 1 u m = µ md ν turbulent 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 /17

10 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

11 Table 19: aminar Entry Region Flow In Circular Tubes Nu D hd k = D/)Re D P r D/)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 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 f/8) 1/ P r /3 1) Nu D hd k lamniar 0.5 P r Re D Above appropriate for both constant T s and constant q s = P e0.87 D lamniar NOT liquid metals P r 5 10 ) q s = constant Re D P e D 10 4 Nu D hd similarly as immediately above = P 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

12 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 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

13 4.3 Concentric Tube Annulus 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) f f = 7. Re 0 D.5D/C) Re D D/C) 1/ Table 5: Correlations for Helically Coiled Tubes Nu D = ) 3 ReD D/C) 1/ ) 3/ ] 1/3 µ a b a = C/D) ) Re D P r b = P r µ s ) P r 1600 D 1/ 1 Re D C /17

14 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

15 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 P r P 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 = ] Applies for all Ra /P r) 9/16 8/7 Better avg. aminar Heat Transfer Nu = Ra 1/4 l /P r) 9/16 ] 4/9 Ra < 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/ < Ra < 10 7 Nu = 0.15Ra 1/ < Ra < Nu = 0.7Ra 1/ < Ra < ong Horizontal Cylinder Assumes isothermal cylinder. The following correlation applies for Ra D < 10 1 Nu D = ] 0.387Ra 1/6 D ] /P r) 9/16 8/7 5.5 Spheres For P r > 0.7 and Ra D < Nu D = Ra 1/4 D /P r) 9/16 ] 4/9 15/17

16 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) )] < S Ra s < < S Ra s < 10 5 S < 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 = 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 Inclined Channels For plates inclined less than 45 degrees from the vertical Nu s = Ra s S/)] 1/4 Fluid properties are evaluated at T = Ts+T This requires Ra s S/) > 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 P /17

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 = P r Ra ) 0.8 H P r Nu = P r Ra < Ra < All properties evaluated at average temp. between hot and cold plates ) < Ra < H 10 P r 10 5 ) < Ra P r 0.+P r 1 H 10 3 P r 10 5 ) < Ra < 10 7 H Nu = 0.4Ra 0.5 P r H 40 1 P r 10 4 Nu = 0.046Ra 1/ < Ra H 40 1 P r Concentric Cylinders For Cylinders we use an effective thermal conductivity k eff k The Rayleigh number uses the corrected length = P r 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 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