BOOKCOMP, Inc. — John Wiley & Sons / Page 512 / 2nd Proofs / Heat Transfer Handbook / Bejan 512 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [512], (74) Lines: 3190 to 3241 ——— 0.49246pt PgVar ——— Normal Page PgEnds: T E X [512], (74) b = t + 1.328 (Re ) 1/2 (6.172) Re = U c ν (6.173) The parameter α is the aspect ratio. α = s/W, where W is the strip width. • Flush-mounted heat sources (Section 6.6.1): Nu = 0.486Pe 0.53 se s x 0.71 k s k f 0.057 (6.174) For the rectangular patch, Nu = 0.60Pe 0.48 c 2 s 2x s + s 0.63 P s 2A 0.18 k sub k f = 1 (6.175) 0.43Pe 0.52 c 2 s 2x s + s 0.70 P s 2A 0.07 k sub k f = 10 (6.176) Here Nu is as defined for the two-dimensional strip, Pe c = U 0 ( s x + se ) α A/P is the source surface area/perimeter ratio. The foregoing correlations are valid for 10 3 ≤ Pe c ≤ 10 5 5 ≤ x s + s /2 s ≤ 150 0.2 ≤ w s s ≤ 5 In the foregoing, w s is the heat source height, P its length, and P w its width. The channel width is W = 12 mm and the height H can vary from 7 to 30 mm. The heat source dimensions covered in the experiments are P = 12 mm, P h = 4, 8, and 12 mm, H −P h = 3, 8, and 12 mm, and P s = 12 mm. • Isolated blocks (Section 6.6.3): Nu = 0.150Re 0.612 (A ∗ ) −0.455 H P 0.727 (6.178) where Nu = ¯ hP /k, ¯ h is the average heat transfer coefficient, and A s is the heat transfer area, A s = 2P h P w + P P w + 2P h P ¯ T s is the average surface temperature and T ∞ is the stream temperature. The Reynolds number is defined as Re = UD h /ν, where U is the average channel BOOKCOMP, Inc. — John Wiley & Sons / Page 513 / 2nd Proofs / Heat Transfer Handbook / Bejan SUMMARY OF HEAT TRANSFER CORRELATIONS 513 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [513], (75) Lines: 3241 to 3305 ——— 0.65236pt PgVar ——— Normal Page * PgEnds: Eject [513], (75) velocity upstream of the heat source, D H is the channel hydraulic diameter at a section unobstructed by the heat source, and ν is the fluid (air) kinematic viscosity. The fraction of the channel open to flow is A ∗ = 1 − P w /W P h /H Equation (6.178) is valid for 1500 ≤ Re ≤ 10 4 0.33 ≤ P h P ≤ 1.00 0.12 ≤ P w W ≤ 1.00 0.583 ≤ H P ≤ 2.50 A realistic error bound is 5%. • Block array (Section 6.6.4): Nu P = 0.348Re 0.6 P (6.182) where the characteristic length for both Nu and Re is the streamwise length of the block, P . • Pin fin heat sinks (Section 6.6.6): Two correlations are given: Nu = 7.12 ×10 −4 C 0.574 ∆p a L 0.223 p d 1.72 (6.185) where in C ∆p = ρL 3 ∆p µ 2 µ is the dynamic viscosity of the air. Equation (6.185) was derived from the data for 5 × 10 6 <C ∆p < 1.5 ×10 8 : Nu = 3.2 ×10 −6 C 0.520 P W a L −0.205 p d 0.89 (6.186) where C P W = ρ 2 LP w µ 3 covers a range of 10 11 to 10 13 . • Single round submerged jet impinging on an isothermal target surface (Section 6.7.2): Nu Pr 0.42 = G D r , H D f 1 (Re) (6.190) BOOKCOMP, Inc. — John Wiley & Sons / Page 514 / 2nd Proofs / Heat Transfer Handbook / Bejan 514 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [514], (76) Lines: 3305 to 3360 ——— 1.58235pt PgVar ——— Normal Page * PgEnds: Eject [514], (76) where f 1 (Re) = 2Re 1/2 (1 + 0.005Re 0.55 ) 1/2 (6.191a) G = D r 1 − 1.1(D/r) 1 + 0.1[(H/D) − 6](D/r) (6.191b) The range of applicability of the foregoing is 2000 ≤ Re ≤ 4 × 10 5 2 ≤ H D ≤ 12 2.5 ≤ r D ≤ 7.50.004 ≤ A r ≤ 0.04 • Single submerged slot jet impinging on an isothermal target surface (Section 6.7.2): Nu Pr 0.42 = 3.06Re m (x/W ) + (H/W ) + 2.78 (6.193) where m = 0.695 − x 2W + H 2W 1.33 + 3.06 −1 (6.194) The range of applicability is 3000 ≤ Re ≤ 9 × 10 4 4 ≤ H W ≤ 20 4 ≤ x W ≤ 50 • Array of round submerged jets impinging on an isothermal target surface (Sec- tion 6.7.2): Nu Pr 0.42 = K A r , H D ,G A r , H D f 2 (Re) (6.195) where f 2 (Re) = 0.5Re 2/3 (6.196a) K = 1 + H/D 0.6/A 1/2 r 6 −0.05 (6.196b) and G is given by eq. (1.191b). The range of validity of the foregoing is 2000 ≤ Re ≤ 10 5 2 ≤ H D ≤ 12 2.5 ≤ r D ≤ 7.50.004 ≤ A r ≤ 0.04 BOOKCOMP, Inc. — John Wiley & Sons / Page 515 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 515 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [515], (77) Lines: 3360 to 3613 ——— 0.43257pt PgVar ——— Normal Page PgEnds: T E X [515], (77) • Array of submerged slot jets impinging on an isothermal target surface (Section 6.7.2): Nu Pr 0.42 = 2 3 A 3/4 r,0 2Re A r /A r,0 + A r,0 /A r 2/3 (6.197) where A r,0 = 60 + 4 h 2W − 2 2 −1/2 (6.198) with a range of validity of 1500 ≤ Re ≤ 4 × 10 4 2 ≤ H W ≤ 80 0.008 ≤ A r ≤ 2.5A r,0 • Single round free surface jet impinging on a square isothermal target surface (Section 6.7.2): Nu Pr 0.4 = C 1 · Re m D i L h D i A r + C 2 · Re n L ∗ L h L ∗ (1 − A r ) (6.202) where A r = πD 2 i 4L 2 h L ∗ = 0.5( √ 2 L h − D i ) + 0.5(L h − D i ) 2 (6.201) These data have been found to be best correlated in the range 1000 ≤ Re Dn ≤ 51,000 for C 1 = 0.516,C 2 = 0.491, and n = 0.532, where the fluid properties are evaluated at the mean of the surface and ambient fluid temperature. NOMENCLATURE Roman Letter Symbols A constant or correlation constant, dimensionless Prandtl number–dependent constant, dimensionless source surface area, m 2 A s surface area, m 2 A T total heat sink surface area, m 2 A ∗ fraction of channel cross section open to flow, m 2 A 1 flow area (aligned tube arrangement), m 2 BOOKCOMP, Inc. — John Wiley & Sons / Page 516 / 2nd Proofs / Heat Transfer Handbook / Bejan 516 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [516], (78) Lines: 3613 to 3613 ——— 0.87755pt PgVar ——— Normal Page PgEnds: T E X [516], (78) A 2 flow area (staggered tube arrangement), m 2 a sphere radius, m fin height, m B correlation constant, dimensionless b parameter defined by eq. (6.172), dimensionless b(x) similarity function, dimensionless C ratio of eddy to turbulent diffusivity, dimensionless constant or correlation constant, dimensionless C ∆p coefficient in eq. (6.185), dimensionless C P w coefficient in eq. (6.186), dimensionless ¯ C coefficient in free stream velocity definition, dimensionless C f friction coefficient, dimensionless c(x) similarity function, dimensionless c p specific heat, J/kg · K D substantial derivative, dimensionless cylinder or sphere diameter, m round jet diameter, m D h hydraulic diameter, m D H channel hydraulic diameter, m d pin diameter, m tube diameter, m d(x) similarity function, dimensionless d h hydraulic diameter, m Ec Eckert number, dimensionless F function, dimensionless ¯ F Prandtl number, dimensionless f friction factor, dimensionless f(η) stream function, dimensionless G parameter defined by eq. (6.131), dimensionless H envelope height, m location outside the boundary layer, m channel height (plate spacing), m h height, m heat transfer coefficient, W/m 2 · K ¯ h mean heat transfer coefficient, W/m 2 · K h ad adiabatic heat transfer coefficient, W/m 2 · K h av average heat transfer coefficient, W/m 2 · K h L heat transfer coefficient for laminar boundary layer, W/m 2 · K h T heat transfer coefficient for turbulent boundary layer, W/m 2 · K i unit vector in x-coordinate direction, dimensionless i step counter, dimensionless row counter, dimensionless (i,j) row and column index, dimensionless j column counter, dimensionless Colburn heat transfer factor, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 517 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 517 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [517], (79) Lines: 3613 to 3613 ——— 0.39821pt PgVar ——— Normal Page PgEnds: T E X [517], (79) j(x) similarity function, dimensionless j unit vector in y-coordinate direction, dimensionless K flow index, dimensionless k thermal conductivity, W/m · K k ∗ plate plate/fluid thermal conductivity ratio, W/m · K k f fluid thermal conductivity, W/m ·K k s mean roughness length scale, dimensionless k sub substrate thermal conductivity, W/m ·K k unit vector in z coordinate direction, dimensionless L envelope length, m length scale factor, m length of heat sink, m length in streamwise direction, m L core core length, m plate length, m cylinder length, m mixing length, m spacing between plates, m 1 leading edge to first block spacing, m 2 last block to trailing edge spacing, m s heat source length, m m exponent, dimensionless N constant, dimensionless N L exponent, dimensionless number of tube rows, dimensionless Nu Nusselt number, dimensionless Nu mean or average Nusselt number, dimensionless Nu D Nusselt number based on diameter, dimensionless Nu D average Nusselt number based on diameter, dimensionless Nu L Nusselt number based on length, dimensionless Nu x Nusselt number based on diameter, dimensionless n exponent, dimensionless number of pins, dimensionless number of plates in stack, dimensionless O order, dimensionless P perimeter, m P h height of block, m heat source height, m P heat source length, m length of block, m P W fanpower,W P w width of block, m heat source width Pe P ´ eclet number, dimensionless Pe c P ´ eclet number used in eqs. (6.175) and (6.176), dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 518 / 2nd Proofs / Heat Transfer Handbook / Bejan 518 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [518], (80) Lines: 3613 to 3613 ——— 0.00577pt PgVar ——— Normal Page PgEnds: T E X [518], (80) Pr Prandtl number, dimensionless Pr T turbulent Prandtl number, dimensionless p pressure, N/m 2 fin pitch, m p + normalized pressure, N/m 2 ¯p mean or average pressure, N/m 2 p ∗ normalized pressure, N/m 2 p m motion pressure, N/m 2 Q total heat generated, W heat sink dissipation, W heater power input, W Q A direct heat transfer component, W Q B conjugate heat transfer component through substrate, W q exponent, dimensionless q A heat dissipation, block A,W q B heat dissipation, block B,W q heat flux, W/m 2 q m mean heat flux, W/m 2 q volumetric heat generation, W/m 3 q (i,j) rate of heat dissipation by block (i,j) in the array, W R thermal resistance, K/W Re Reynolds number, dimensionless Re ∗ b Reynolds number defined by eq. (6.171), dimensionless Re D Reynolds number based on diameter, dimensionless Re D,max Reynolds number at maximum flow, dimensionless Re k roughness Reynolds number, dimensionless Re L Reynolds number based on length, dimensionless Re Reynolds number defined by eq. (6.173), dimensionless Re P h Reynolds number based on P h defined in Section 6.6.2, dimensionless Re P Reynolds number based on P defined in Section 6.6.2, dimensionless Re T critical Reynolds number, dimensionless Re x Reynolds number based on x, dimensionless Re ∗ transition Reynolds number, dimensionless r boundary layer ratio, dimensionless r c recovery factor, dimensionless r 0 distance from axis to surface, m S jet spacing, m S D diagonal tube spacing, m S L longitudinal tube spacing, m S T transverse tube spacing, m St Stanton number, dimensionless St k roughness Stanton number, dimensionless BOOKCOMP, Inc. — John Wiley & Sons / Page 519 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 519 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [519], (81) Lines: 3613 to 3613 ——— 0.92038pt PgVar ——— Normal Page PgEnds: T E X [519], (81) s strip spacing, m clear space between blocks, m s x x-coordinate distance, m s z distance to bounding surface, m T temperature, K ¯ T average or mean temperature, K ¯ T s average surface temperature, K T air,B temperature of air at block B,K T b temperature in buffer region, K temperature at bottom surface of heat sink, K T max maximum surface temperature, K T ref reference temperature, K T s surface temperature, K T s,B surface temperature of block B,K T ∗ normalized temperature in eq. (6.8), dimensionless T + b normalized buffer temperature, K T 0 free stream temperature, K air temperature at front of heat sink, K T ∞ ambient temperature, K t time, s plate thickness, m nondimensional substrate thickness, dimensionless t ∗ normalized time, s t 0 time reference, s U overall heat transfer coefficient, W/m 2 · K velocity scale factor, m/s free stream velocity, m/s U ∞ velocity in undisturbed flow, m/s U 0 average velocity in unobstructed channel, m/s U c core velocity, m/s ux-coordinate velocity, m/s u + normalized x-coordinate velocity, m/s u + ∞ normalized free stream x-coordinate velocity, m/s u o free stream x-coordinate velocity, m/s ¯u mean or average x-coordinate velocity, m/s u ∗ normalized x-coordinate velocity, m/s V velocity, m/s V max maximum velocity, m/s V velocity vector, m/s vy-coordinate velocity, m/s v 0 free stream y-coordinate velocity, m/s v ∗ friction velocity in turbulent flow, m/s v + normalized y-coordinate velocity, m/s ¯v mean or average y-coordinate velocity, m/s BOOKCOMP, Inc. — John Wiley & Sons / Page 520 / 2nd Proofs / Heat Transfer Handbook / Bejan 520 FORCED CONVECTION: EXTERNAL FLOWS 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [520], (82) Lines: 3613 to 3655 ——— 0.20847pt PgVar ——— Normal Page PgEnds: T E X [520], (82) W envelope width, m channel width, m w width, m z-coordinate velocity, m/s slot jet width, m w ∗ normalized, z-coordinate velocity, m/s X body force, N x 0 x coordinate, m x unheated starting length, m x ∗ normalized x-coordinate velocity, m/s x s leading edge to heat source distance, m Y body force, N y ∗ normalized y-coordinate velocity, m/s z ∗ normalized z-coordinate velocity, m/s ∆p pressure difference, N/m 2 ∆p core core pressure difference, N/m 2 ∆T temperature difference, K Greek Letter Symbols α thermal diffusivity, m 2 /s aspect ratio, dimensionless β volumetric expansion coefficient, K −1 wedge angle, rad constant, pressure difference, N/m 2 Γ gamma function, dimensionless ∆ change in, dimensionless δ hydrodynamic boundary layer, m δ 2 momentum thickness, m δ c conduction thickness, m δ T thermal boundary layer, m eddy viscosity, m 2 /s H eddy diffusivity, m 2 /s η similarity variable, dimensionless similarity function, dimensionless η B Blasius similarity variable, dimensionless η δ thickness of boundary layer, dimensionless θ normalized temperature, dimensionless angle, rad θ B/A effect of heat dissipation from block B on block A, K/W θ B/(i,j) contribution of all blocks upstream of block B, K/W θ hot hot spot temperature, dimensionless κ von K ´ arm ´ an constant, dimensionless µ dynamic viscosity, kg/m ·s µ s dynamic viscosity at surface or wall temperature, kg/m · s ν kinematic viscosity, m 2 /s BOOKCOMP, Inc. — John Wiley & Sons / Page 521 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 521 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [521], (83) Lines: 3655 to 3700 ——— 0.20847pt PgVar ——— Normal Page PgEnds: T E X [521], (83) ξ dummy variable, dimensionless ρ fluid density, kg/m 3 τ shear stress, N/m 2 τ b mean shear stress, N/m 2 τ o free stream shear stress, N/m 2 τ T turbulent shear stress, N/m 2 Φ viscous dissipation, s −2 φ(x,y) stream function, dimensionless φ(η) similarity function, dimensionless ψ(x,y) stream function, dimensionless Roman Letter Subscripts A block designator direct heat transfer component AW adiabatic wall ad, B adiabatic heat transfer coefficient on block B air, b air temperature at block B av average B pertaining to Blasius conjugate heat transfer component block designator B/A effect on block B by dissipation from block A b/(i,j ) contribution of upstream block b bottom of heat sink c conduction thickness recovery factor core core D diameter f fluid friction H homogeneous art of solution h height hot max (i,j) row and column index L length length opt optimum P particular part of solution P w fan power p constant pressure plate plate ref reference s surface se heat source sub substrate . m heat transfer coefficient, W/m 2 · K ¯ h mean heat transfer coefficient, W/m 2 · K h ad adiabatic heat transfer coefficient, W/m 2 · K h av average heat transfer coefficient, W/m 2 · K h L heat transfer. N/m 2 p m motion pressure, N/m 2 Q total heat generated, W heat sink dissipation, W heater power input, W Q A direct heat transfer component, W Q B conjugate heat transfer component through substrate,. dimensionless q A heat dissipation, block A,W q B heat dissipation, block B,W q heat flux, W/m 2 q m mean heat flux, W/m 2 q volumetric heat generation, W/m 3 q (i,j) rate of heat dissipation