BOOKCOMP, Inc. — John Wiley & Sons / Page 472 / 2nd Proofs / Heat Transfer Handbook / Bejan 472 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 [472], (34) Lines: 1539 to 1601 ——— 7.28726pt PgVar ——— Normal Page PgEnds: T E X [472], (34) τ = τ m + τ T = ρ(ν + ) ∂ ¯u ∂y (6.86) q = q m + q T =−ρc p (α + H ) ∂ ¯ T ∂y (6.87) These can be integrated to yield ¯u = y 0 τ ρ(ν + ) dy (6.88) T 0 − ¯ T = y 0 q ρc p (α + H ) dy (6.89) 6.4.12 Algebraic Turbulence Models The simplest class of turbulence closure models is the zero-equation model or first- order closure model. Prandtl’s mixing length model is an example of this class where the eddy diffusivities and H are modeled in terms of the gradients of the mean flow: = 2 ∂ ¯u ∂y and H = C 2 ∂ ¯u ∂y (6.90) where is the mixing length. Here C is found from Pr T : C = H = 1 Pr T where Pr T is the turbulent Prandtl number. 6.4.13 Near-Wall Region in Turbulent Flow The mean momentum equation can be written near the wall as ρ ¯u ∂ ¯u ∂x + ρ ¯v ∂ ¯u ∂y − ∂τ ∂y + d ¯p dx = 0 (6.91) In a region close to the wall, the approximations ρ ¯u ∂ ¯u ∂x ≈ 0 (6.92a) ¯u =¯u(y) (6.92b) and the normal velocity at the wall, ¯v = v 0 (6.92c) BOOKCOMP, Inc. — John Wiley & Sons / Page 473 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 473 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 [473], (35) Lines: 1601 to 1659 ——— 0.94249pt PgVar ——— Normal Page * PgEnds: Eject [473], (35) can be made. With these, the momentum equation becomes ρv 0 ∂ ¯u ∂y − ∂τ ∂y + d ¯p dx = 0 which can be integrated using τ = τ 0 and ¯u = 0aty = 0: τ τ 0 = 1 + ρv 0 ¯u τ 0 + d ¯p dx y τ 0 (6.93) Near the wall, a friction velocity v ∗ can be defined as τ 0 ρ = C f U 2 2 = (v ∗ ) 2 which yields v ∗ = τ 0 ρ 1/2 (6.94) Moreover, normalized variables can be defined near the wall: u + = ¯u v ∗ = ¯u/U C f /2 (6.95a) v + = yv ∗ ν (6.95b) v + 0 = v 0 v ∗ (6.95c) p + = µ(d ¯p/dx) ρ 1/2 τ 3/2 0 (6.95d) and substitution of eqs. (6.95) into eq. (6.93) yields τ τ 0 = 1 + v + u + + p + y + (6.96) In the absence of pressure gradient and transpiration (p + = 0,v + 0 = 0), τ τ 0 = 1orτ 0 = ρ(ν + ) d ¯u dy (6.97) In a region very close to the wall (called the viscous sublayer), ν and eq. (6.97) can be integrated from the wall to a nearby location (say, 0 ≤ y + ≤ 5) in the flow ¯u 0 d ¯u = τ 0 µ y 0 dy or u + = y + (6.98) BOOKCOMP, Inc. — John Wiley & Sons / Page 474 / 2nd Proofs / Heat Transfer Handbook / Bejan 474 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 [474], (36) Lines: 1659 to 1704 ——— 0.68515pt PgVar ——— Long Page PgEnds: T E X [474], (36) In a region farther out from the wall, turbulence effects become important. However, the near-wall region still has effectively constant total shear stress and total heat flux: τ = τ 0 + ρ(ν + ) ∂ ¯u ∂y = ρ ν + l 2 ∂ ¯u ∂y ∂ ¯u ∂y = ρ ν + κ 2 y 2 ∂ ¯u ∂y ∂ ¯u ∂y (6.99) where the mixing length near the wall is taken as = κy, with κ = 0.40 being the von K ´ arm ´ an constant. In the fully turbulent outer region, ν , resulting in τ 0 = ρκ 2 y 2 ∂ ¯u ∂y 2 or κy ∂ ¯u ∂y = τ 0 ρ 1/2 = v ∗ (6.100) This can be expressed as κy + ∂u + ∂y + = 1 which can be integrated to yield the law of the wall: u + = 1 κ ln y + + C (6.101) With κ = 0.4 and C = 5.5, this describes the data reasonably well for y + > 30, as seen in Fig. 6.14. Minor variations in the κ and C values are found in the literature, as indicated in Fig. 6.14. In the range 5 <y + < 30, the buffer region, both ν Figure 6.14 Velocity profiles in turbulent flow past a flat surface in the wall coordinates. (Data from the literature, as reported by Kays and Crawford, 1993.) BOOKCOMP, Inc. — John Wiley & Sons / Page 475 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 475 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 [475], (37) Lines: 1704 to 1765 ——— 2.31464pt PgVar ——— Long Page * PgEnds: Eject [475], (37) and are important and the experimental data from various investigators can be approximated by u + = 5lny + − 3.05 6.4.14 Analogy Solutions for Boundary Layer Flow Consider the apparent shear stress and heat flux in turbulent flat plate boundary layer flow: τ = ρ(ν + ) ∂ ¯u ∂y and ∂ ¯u ∂y = τ ρ(ν + ) (6.102) q =−ρc p (α + H ) ∂ ¯ T ∂y and ∂ ¯ T ∂y =− q ρc p (α + H ) (6.103) These expressions can be integrated from the wall to a location in the free stream, u = 0 τ ρ(ν + ) dy (6.104) T 0 − T = 0 q ρc p (α + H ) dy (6.105) Under the assumptions that Pr = Pr T = 1 and q q 0 = τ τ 0 then T 0 − T = 0 q ρc p (ν + ) dy = q 0 τ 0 0 τ ρc p (ν + ) dy (6.106) Upon division by the velocity expression T 0 − T u = q 0 c p τ 0 then q 0 T 0 − T x k = τ 0 ρu 2 ρu 1 x µ µc p k or Nu x = C f 2 Re x · Pr (6.107) For moderate Re x C f = 0.058 Re 0.2 x BOOKCOMP, Inc. — John Wiley & Sons / Page 476 / 2nd Proofs / Heat Transfer Handbook / Bejan 476 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 [476], (38) Lines: 1765 to 1821 ——— 7.43932pt PgVar ——— Normal Page * PgEnds: Eject [476], (38) and for Pr = Pr T = 1 and no pressure gradient, Nu x = 0.0296Re 0.80 x (6.108) Mixed Boundary Conditions Often, the boundary layer may start as laminar, undergo transition, and become turbulent along the plate length. The average heat transfer coefficient in such a case can be determined by assuming an abrupt transition at a location x T : ¯ h = 1 L x T 0 h L dx + L x T h T dx (6.109) where h L and h T are the local heat transfer coefficient variations for the laminar and turbulent boundary layers, respectively. A critical Reynolds number Re T can be selected and eq. (6.109) can be normalized to obtain the average Nusselt number Nu L = 0.664Re 0.50 T + 0.036 Re 0.8 L − Re 0.8 T (6.110) Three-Layer Model for a “Physical Situation” Considering the total shear stress and heat flux in eqs. (6.102) and (6.103), division and introduction of the turbulent scales leads to τ τ 0 = 1 + ν ∂u + ∂y + (6.111a) q q 0 = 1 Pr + /ν Pr T ∂T + ∂y + (6.111b) The turbulent boundary layer may be considered to be divided into an inner region, a buffer region, and an outer region. For the inner region (0 ≤ y + < 5), = 0 and q = q 0 , which results in 1 = 1 Pr ∂T + ∂y + or T + = Pr · y + (6.112) At the outer edge of the inner layer, T + s = 5Pr. For the buffer region (5 ≤ y + < 30), τ = τ 0 q = q 0 u + = 5 + 5ln y + 5 (6.113a) or ∂u + ∂y + = 5 y + (6.113b) BOOKCOMP, Inc. — John Wiley & Sons / Page 477 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 477 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 [477], (39) Lines: 1821 to 1886 ——— 0.88054pt PgVar ——— Normal Page * PgEnds: Eject [477], (39) Substituting eq. (6.113a) into eqs. (6.111) provides 1 = 1 + ν 5 y + (6.114a) or ν = y + 5 − 1 (6.114b) With eq. (6.114b) put into eq. (6.112), 1 = 1 Pr + y + − 5 5Pr T ∂T + ∂y + (6.115) Then integration across the buffer region gives T + − T + s = y + 5 dy + 1/Pr + (y + − 5)/5Pr T (6.116) or T + − T + s = 5Pr T ln 1 + Pr Pr T y + 5 − 1 (6.117) This may be applied across the entire buffer region: T + b − T + s = 5Pr T ln 1 + Pr Pr T 30 5 − 1 (6.118) For the outer region (y + > 30), where ν and H ν, τ τ 0 = ν ∂u + ∂y + (6.119) q q 0 = /ν Pr ∂T + ∂y + (6.120) Similar distributions of shear stress and heat transfer rate in the outer region can be assumed so that τ τ b = q q b (6.121) Because τ and q are constant across the inner two layers, τ b = τ 0 and q = q 0 . Hence τ τ 0 = q q 0 (6.122) BOOKCOMP, Inc. — John Wiley & Sons / Page 478 / 2nd Proofs / Heat Transfer Handbook / Bejan 478 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 [478], (40) Lines: 1886 to 1969 ——— 0.84541pt PgVar ——— Long Page * PgEnds: Eject [478], (40) With eqs. (6.121) and (6.122) in eq. (6.120), ν ∂u + ∂y + = νPr T ∂T + ∂y + and ∂T + ∂y + = Pr T ∂u + ∂y + (6.123) Integrating across the outer layer yields T + 1 − T + b = Pr T (u + 1 − u + b ) (6.124) where T + 1 and u + 1 are in the free stream and u + b = 5 + 5ln6 so that T + 1 − T + b = Pr T u + 1 − 5(1 + ln 6) (6.125) Addition of the temperature profiles across the three layers gives T + 1 = 5Pr + 5Pr T ln Pr T + 5Pr 6Pr T + Pr T u + 1 − 5 (6.126) and because T + = ρc p (T 0 − ¯ T) q 0 v ∗ = ρc p (T 0 − ¯ T) q 0 τ 0 ρ 1/2 = Pr · Re x Nu x C f 2 1/2 (6.127) the Nusselt number becomes Nu x = Re x (C f /2) 1/2 5 + 5 Pr T Pr ln Pr T + 5Pr 6Pr T + Pr T Pr (u + 1 − 5) (6.128) This yields the final expression with St = Nu x /Re x · Pr: St x = (C f /2) 1/2 (C f /2) 1/2 {5Pr + 5Pr T ln[(Pr T + 5Pr)/6Pr T ] − 5Pr T }+Pr T (6.129) For C f = 0.058 Re 0.20 x eq. (6.129) can be written as Nu x = 0.029Re 0.8 x G (6.130) where the parameter G is BOOKCOMP, Inc. — John Wiley & Sons / Page 479 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 479 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 [479], (41) Lines: 1969 to 2027 ——— 3.1273pt PgVar ——— Long Page * PgEnds: Eject [479], (41) G = Pr (0.029/Re .20 x ) 1/2 {5Pr + 5ln[(1 + 5Pr)/6] − 5}+1 (6.131) As pointed out by Oosthuizen and Naylor (1999), eq. (6.131) is based on Pr T = 1 but does apply for various Pr. Flat Plate with an Unheated Starting Length in Turbulent Flow The inte- gral forms of the momentum and energy equations can also be developed for turbulent flow by integrating the mean flow equations across the boundary layer from the sur- face to a location H outside the boundary layer. The integral momentum equation can be written as d dx H 0 ¯u( ¯u − U)dy =− τ 0 ρ =−U 2 dδ 2 dx =−(v ∗ ) 2 (6.132) where the momentum thickness δ 2 is δ 2 = H 0 ¯u U 1 − ¯u U dy (6.133) Assume that both Pr and Pr T are equal to unity and that the mean profiles for the mean velocity and temperature in the velocity (δ) and thermal (δ T ) boundary layers are given by ¯u U = y δ 1/7 and T 0 − ¯ T t 0 − T ∞ = y δ T 1/7 (6.134) Then from the integral momentum equation τ τ 0 = 1 − y δ 1/7 (6.135) and because the total heat flux at any point in the flow is ν + = α + H = τ/ρ ∂ ¯u/∂y = 7ν C f 2 δ x Re x 1 − y δ 9/7 y δ 6/7 (6.136) the total heat flux at any point in the flow is q =−ρc p (α + H ) ∂ ¯ T ∂y (6.137) For the assumed profiles this can be written as q ρc p U(T 0 − T ∞ ) = C f 2 1 − y δ 9/7 y δ 6/7 (6.138) BOOKCOMP, Inc. — John Wiley & Sons / Page 480 / 2nd Proofs / Heat Transfer Handbook / Bejan 480 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 [480], (42) Lines: 2027 to 2074 ——— 1.5713pt PgVar ——— Normal Page PgEnds: T E X [480], (42) The wall heat flux can be determined from eq. (6.138). This expression is substi- tuted on the right-hand side of the integral energy equation. The ¯u and ¯ T profiles and the δ as a function of x relations are substituted on the left side and δ T is solved for as a function of x. The final result for the Stanton number St is St = C f 2 1 − x 0 x 9/10 −1/9 (6.139) With the same Prandtl number correction as in the case of the fully heated plate, St · Pr 0.40 = 0.0287Re −0.2 x 1 − x 0 x 9/10 −1/9 (6.140) Just as for laminar flow, the result of eq. (6.140) can be generalized to an arbitrarily varying wall temperature through superposition (Kays and Crawford, 1980). Arbitrarily Varying Heat Flux Based on eq. (6.134), which is Prandtl’s 1 7 power law for thermal boundary layers on smooth surfaces with uniform specified surface heat flux, a relationship that is nearly identical to eq. (6.140) for uniform surface temperature is recommended: St · Pr 0.40 = 0.03Re −0.2 x (6.141) For a surface with an arbitrarily specified heat flux distribution and an unheated section, the temperature rise can be determined from T 0 (x) −T ∞ = x 0 =x x 0 =0 Γ(x 0 ,x)q (x 0 )dx 0 (6.142) where Γ(x 0 ,x) = 9 10 Pr −0.60 · Re −0.80 x Γ 1 9 Γ 8 9 (0.0287k) 1 − x 0 x 9/10 −8/9 (6.143) Turbulent Prandtl Number An assumption of Pr T = 1 has been made in the foregoing analyses. The turbulent Prandtl number is determined experimentally using Pr T = u v (∂ ¯ T/∂y) v T (∂ ¯u/∂y) (6.144) Kays and Crawford (1993) have pointed out that, in general, it is difficult to measure all four quantities accurately. With regard to the mean velocity data, mean temperature data in the logarithmic region show straight-line behavior when plotted in the wall coordinates. The slope of this line, relative to that of the velocity profile, provides Pr T . Data for air and water reveal a Pr T range from 0.7 to 0.9. No noticeable effect BOOKCOMP, Inc. — John Wiley & Sons / Page 481 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 481 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 [481], (43) Lines: 2074 to 2114 ——— 3.49532pt PgVar ——— Normal Page * PgEnds: Eject [481], (43) due to surface roughness is noted, but at very low Pr, a Prandtl number effect on Pr T is observed. This is believed to be due to the higher thermal conductivity based on existing analyses. At higher Pr, there is no effect on Pr T . The Pr T values are found to be constant in the “law of the wall” region but are higher in the sublayer, and the wall value of Pr T approaches 1.09 regardless of Pr. 6.4.15 Surface Roughness Effect If the size of the roughness elements is represented by a mean length scale k s ,a roughness Reynolds number may be defined as Re k = k s (v ∗ /ν). Three regimes of roughness may then be prescribed: Smooth Re k < 5 Transitional rough 5 < Re k < 70 Fully rough Re k > 70 Under the fully rough regime, the friction coefficient becomes independent of viscosity and Re k . The role of the roughness is to destabilize the sublayer. For Re k > 70, the sublayer disappears and the shear stress is transmitted to the wall by pressure drag on the roughness elements. The mixing length near a rough surface is given as = κ(y+δy o ), which yields a finite eddy diffusivity at y = 0. In the wall coordinates, based on experimental data, δy + 0 = δy 0 v ∗ ν = 0.031Re k (6.145) The law of the wall for a fully rough surface is du + dy + = 1 κ(y + + δy + 0 ) (6.146) Because no sublayer exists for fully rough surface, eq. (6.146) may be integrated from 0toy + to provide u + = 1 κ ln y + + 1 κ ln 32.6 Re k (6.147) The friction coefficient is calculated by evaluating u + ∞ while including the additive effect of the outer wake. The latter results in a displacement by 2.3. The result is u + ∞ = 1 (C f /2) 1/2 = 1 κ ln 848 k s (6.148) The heat transfer down to the plane of the roughness elements is by eddy con- ductivity, but the final transfer to the surface is by molecular conduction through the almost stagnant fluid in the roughness cavities. The law of the wall is written as . v 0 (6.92c) BOOKCOMP, Inc. — John Wiley & Sons / Page 473 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 473 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 [473],. Crawford, 1993.) BOOKCOMP, Inc. — John Wiley & Sons / Page 475 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 475 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 [475],. 5ln y + 5 (6.113a) or ∂u + ∂y + = 5 y + (6.113b) BOOKCOMP, Inc. — John Wiley & Sons / Page 477 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 477 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 [477],