BOOKCOMP, Inc. — John Wiley & Sons / Page 452 / 2nd Proofs / Heat Transfer Handbook / Bejan 452 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 [452], (14) Lines: 509 to 566 ——— -2.12363pt PgVar ——— Long Page PgEnds: T E X [452], (14) u ∗ ∂T ∗ ∂x ∗ + δ δ T v ∗ ∂T ∗ ∂y ∗ T = 1 Pr δ δ T 2 ∂ 2 T ∗ ∂y ∗2 T + 2Ec · Φ ∗ + 2βT · Ec · u ∗ dp ∗ dx ∗ + q L ρc p U ∆T (6.21) The corresponding dimensional form of eq. (6.21) is u ∂T ∂x + v ∂T ∂y = α ∂ 2 T ∂y 2 + βT ρc p u dp dx + µ ρc p ∂u ∂y 2 + q ρc p (6.22) Equation (6.20) can be used to demonstrate that when Pr 1, δ T δ = O 1 Pr 1/2 and when Pr 1, δ T δ = O 1 Pr 1/3 6.4.2 Similarity Transformation Technique for Laminar Boundary Layer Flow Following the simplifications of eqs. (6.19)–(6.22), the two-dimensional steady-state boundary layer equations are: ∂u ∂x + ∂v ∂y = 0 (6.23) u ∂u ∂x + v ∂u ∂y =− 1 ρ ∂p ∂x + ν ∂ 2 u ∂y 2 = UU x + ν ∂ 2 u ∂y 2 (6.24) u ∂T ∂x + v ∂T ∂y = α ∂T 2 ∂y 2 + βTu ∂p ∂x + µ ρc p ∂u ∂y 2 + q ρc p (6.25) The boundary conditions for an impermeable surface are u(x,0) = v(x,0) = 0 (6.26a) T(x,0) = T 0 (x) (6.26b) u(x, ∞) = U(x) (6.26c) T(x,∞) = T ∞ (x) (6.26d) Equations (6.23)–(6.25) constitute a set of nonlinear partial differential equations. Under certain conditions similarity solutions can be found that allow conversion of this set into ordinary differential equations. The concept of similarity means that BOOKCOMP, Inc. — John Wiley & Sons / Page 453 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 453 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 [453], (15) Lines: 566 to 615 ——— 2.14024pt PgVar ——— Long Page PgEnds: T E X [453], (15) certain features (e.g., velocity profiles) are geometrically similar. Analytically, this amounts to combining the x and y spatial dependence on a single independent vari- able η. The velocity components u(x,y) and v(x,y) are expressed by a single nondi- mensional stram function f(η), and the temperature T(x,y)into a nondimensional temperature φ(η). Similarity for boundary layer flow follows from the observation that while the boundary layer thickness at each downstream location x is different, a scaled normal distance η can be employed as a universal length scale. Presence of natural length scales (such as a finite-length, plate, cylinder, or sphere) generally precludes the finding of similarity solutions. Using the scaled distance, the similarity procedure finds the appropriate normalized stream and temperature functions that are also valid at all locations. Following Gebhart (1980), the similarity variables are defined as η(x,y) = yb(x) (6.27) f(η) = ψ(x,y) vc(x) (6.28) φ(η) = T(x,y)− T ∞ (x) T 0 (x) − T ∞ (x) (6.29) where the allowable forms of b(x) and c(x) (defined later) and d(x) = T 0 (x) − T ∞ (x) j(x) = T ∞ (x) − T ref need to be determined. The transformed governing equations in terms of the foregoing normalized variables are f (η) + 1 b(x) dc(x) dx f(η)f (η) − 1 b(x) dc(x) dx + c(x) b(x) db(x) dx [f (η)] 2 − 1 ρv 2 c(x)[(b(x)] 3 dp dx = 0 (6.30) φ (η) Pr + 1 b(x) dc(x) dx f(η)f (η) − c(x) b(x)d(x) dd(x) dx f (η)φ(η) − c(x) b(x)d(x) dj (x) dx f (η) + βT ρc p c(x) b(x)d(x) dp dx f (η) + ν 2 c p [c(x)b(x)] 2 d(x) [f (η)] 2 + 1 d(x)[b(x)] 2 q kPr = 0 (6.31) The boundary conditions are f (0) = f(0) = 1 − φ(0) = φ(∞) = 0 BOOKCOMP, Inc. — John Wiley & Sons / Page 454 / 2nd Proofs / Heat Transfer Handbook / Bejan 454 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 [454], (16) Lines: 615 to 681 ——— 0.08028pt PgVar ——— Normal Page * PgEnds: Eject [454], (16) and it is also noted that dp dx =−ρU dU dx For the momentum equation to be entirely a function of the independent variable η, 1 b(x) dc(x) dx = C 1 c(x) [b(x)] 2 db(x) dx = C 2 This results in choices for the constants, C 1 and C 2 : c(x) = k e kx (C 1 = C 2 ) k x q (C 1 = C 2 ) with C 1 and C 2 related by C 2 = q − 1 q C 1 For the pressure gradient term to be independent of x, the free stream velocity must be U(x) ∝ x 2q−1 Also known as Falkner–Skan flow, this form arises in the flow past a wedge with an included angle βπ as seen in Fig. 6.6. In this case, U(x) = ¯ Cx m where m = β 2 − β as indicated in potential flow theory. From the similarity requirement, the exponent q becomes q = m + 1 2 = 1 2 − β Then the pressure gradient term in eq. (6.30) becomes − 1 ρν 2 1 c(x)[b(x)] 3 dp dx = (C 1 − C 2 ) 3 m ¯ C 2 ν 2 k 4 x 2m−1 x 4q−3 = β BOOKCOMP, Inc. — John Wiley & Sons / Page 455 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 455 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 [455], (17) Lines: 681 to 735 ——— 1.99225pt PgVar ——— Normal Page * PgEnds: Eject [455], (17) The arbitrary constants C 1 and C 2 may be chosen, without loss of generality, as C 1 = 1 and C 2 = β − 1 This results in c(x) = 2 m + 1 Re x 1/2 b(x) = 1 x m + 1 2 Re x 1/2 η = y x m + 1 2 Re x 1/2 ψ(x,y) = νf(η) 2 m + 1 Re x 1/2 and for this choice of constants, the Falkner–Skan momentum equation and boundary conditions become f (η) + f(η)f (η) + 1 − f (η) 2 β = 0 f (η = 0) = f(η = 0) = 1 − f (η =∞) = 0 The often used (and much older) Blasius (1908) variables for flow past a flat plate (β = 0) are related to the Falkner–Skan variables η and f(η) as η B = 2 1/2 η and f(η B ) = 2 1/2 f(η) For similarity to hold, the energy equation must satisfy the conditions c(x) b(x)d(x) dd(x) dx = C 5 (6.32a) ν 2 [c(x)b(x)] 2 c p d(x) = K 3 [c(x)b(x)] 2 d(x) = K 3 C 6 (6.32b) c(x) b(x)d(x) dj (x) dx = C 7 (6.32c) βT ρc p c(x) b(x)d(x) dp dx = K 4 c(x) b(x)d(x) dp dx = K 4 C 10 (6.32d) 1 d(x)[b(x)] 2 1 Pr q k = F 1 (η) (6.32e) BOOKCOMP, Inc. — John Wiley & Sons / Page 456 / 2nd Proofs / Heat Transfer Handbook / Bejan 456 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 [456], (18) Lines: 735 to 799 ——— 0.86714pt PgVar ——— Short Page * PgEnds: Eject [456], (18) 6.4.3 Similarity Solutions for the Flat Plate at Uniform Temperature (m = 0) For the case of the flat plate, the similarity equations and boundary conditions reduce to f (η) + f(η)f (η) = 0 φ (η) + Pr · f(η)φ (η) = 0 (6.33) and 1 − φ(0) = φ(∞) = 0 and f(0) = f (0) = 1 − f (∞) = 0 (6.34) Both the momentum and energy equations are ordinary differential equations in the form of two-point boundary value problems. The momentum equation is solved first because it is uncoupled from the energy equation. The velocity field is then substituted into the energy equation to obtain the temperature field and heat transfer characteristics. The wall heat flux is obtained as q (x) = h x (T 0 − T ∞ ) =−k ∂T ∂y y=0 =−k(T 0 − T ∞ )φ (0) ∂η ∂y =−k(T 0 − T ∞ )φ (0) 1 x · Re 1/2 x (6.35) This results in the local Nusselt number Nu x = h x x k = −φ (0) √ 2 Re 1/2 x = ¯ F(Pr)Re 1/2 x where ¯ F(Pr) is determined numerically and near Pr ≈ 1 is well approximated by 0.332Pr 1/3 so that, Nu x = 0.332Re 1/2 x · Pr 1/3 (6.36) the surface-averaged heat transfer coefficient is determined as: ¯ h = 1 A s hdA s = 1 L L 0 h x dx = 2h x=L (6.37) 6.4.4 Similarity Solutions for a Wedge (m = 0) For a wedge at a uniform surface temperature, the expressions for the surface heat flux and the Nusselt number are BOOKCOMP, Inc. — John Wiley & Sons / Page 457 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 457 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 [457], (19) Lines: 799 to 861 ——— 0.3433pt PgVar ——— Short Page * PgEnds: Eject [457], (19) q (x) =−k(T 0 − T ∞ )φ (0) 1 x m + 1 2 Re x 1/2 (6.38) Nu x = h x x k = −φ (0) √ 2 [(m + 1)Re x ] 1/2 = ¯ F (m, Pr)Re 1/2 x (6.39) For a wedge with a spatially varying surface temperature, eq. (6.32a) C 5 = c(x) b(x)d(x) dd(x) dx yields 1 d(x) dd(x) dx = C 5 b(x) c(x) = C 5 m + 1 2x = n x where n ≡ C 5 m + 1 2 is a constant. This yields d(x) = [T 0 (x) − T ∞ ] = Nx n where N is a constant arising from the integration for the surface temperature varia- tion. The energy equation is transformed to φ (η) + Pr f(η)φ (η) − 2n m + 1 f (η)φ(η) = 0 The resulting expressions for the heat flux and the local Nusselt number are q (x) =−kφ (0)N (m + 1) ¯ C 2ν 1/2 x (2n+m−1)/2 (6.40) Nu x =−φ (0) m + 1 2 Re x 1/2 = ¯ F(Pr,m,n)Re 1/2 x (6.41) In the range of 0.70 ≤ Pr ≤ 10, Zhukauskas (1972, 1987) reports the correlation for the data computed by Eckert: Nu Re 1/2 x = 0.56A (2 − β) 1/2 (6.42) where BOOKCOMP, Inc. — John Wiley & Sons / Page 458 / 2nd Proofs / Heat Transfer Handbook / Bejan 458 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 [458], (20) Lines: 861 to 895 ——— 0.24107pt PgVar ——— Normal Page * PgEnds: Eject [458], (20) β = 2m m + 1 and A = (β + 0.20) 0.11 · Pr 0.333+0.067β−0.026β 2 For the thermal boundary layer thickness to increase with x, two special cases of the foregoing solution corresponding to a flat plate (m = 0) are of interest (Fig. 6.7). These correspond respectively to a uniform heat flux surface and a line heat source at x = 0 (a line plume). The heat flux can be written as q (x) =−kφ (0)N U ν 1/2 x (2n−1)/2 For the first condition (Fig. 6.7a), n = 1 2 . For the second condition (Fig. 6.7b), the total energy convected by the flow per unit length of the source is written as q (x) = ∞ 0 ρc p u(T − T ∞ )dy = νρc p c(x)d(x) ∞ 0 f (η)φ(η)dη ∝ x (2n+1)/2 (6.43) For the convected energy to remain invariant with x, n must take on the value n =− 1 2 . Wedge Flow Limits With regard to wedge flow limits, numerical solutions to the Falkner–Skan equations have been obtained for −0.0904 ≤ m ≤∞, where the lower limit is set by the onset of boundary layer separation. In addition, the hydrodynamic boundary layer thickness is Figure 6.7 Two important cases for boundary layer flow at uniform free stream velocity. BOOKCOMP, Inc. — John Wiley & Sons / Page 459 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 459 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 [459], (21) Lines: 895 to 945 ——— 0.66211pt PgVar ——— Normal Page PgEnds: T E X [459], (21) δ(x) = η δ 2ν (m + 1) ¯ C 1/2 x (1−m)/2 where η δ , the nondimensional thickness of the boundary layer, is bound at x = 0 only for m ≤ 1. This requires that 1 ≥ m ≥−0.0904. Additionally, the total energy convected by the flow per unit width normal to the plane of flow is given by q (x) = νρc p c(x)d(x) ∞ 0 f (η)φ(η)dη ∝ Nx (2n+m+1)/2 This provides the condition 2n + m + 1 2 ≥ 0orn ≥− m + 1 2 6.4.5 Prandtl Number Effect Consider the case of n = m = 0 first. In the limiting cases of Pr 1 and Pr 1 (Fig. 6.8), the solution of the momentum equation can be approximated in closed form. Subsequently, the energy equation can be solved. For Pr 1, the velocity components u and U are approximately equal throughout the thermal boundary layer. This results in f (η) ≈ 1orf(η) ≈ η + K. For an impermeable wall, K = 0 and the energy equation simplifies to φ (η) + Pr · ηφ (η) = 0 This can be integrated to yield φ (η) = e −η 2 1 +C where η 2 1 = η 2 · Pr 2 Figure 6.8 Laminar flat plate boundary layer flow at limiting Prandtl numbers. BOOKCOMP, Inc. — John Wiley & Sons / Page 460 / 2nd Proofs / Heat Transfer Handbook / Bejan 460 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 [460], (22) Lines: 945 to 1016 ——— 5.77745pt PgVar ——— Normal Page PgEnds: T E X [460], (22) and with the boundary conditions 1 = φ(0) = φ(∞) = 0 another integration provides φ(η) = 1 − erf(η 1 ) where erf(η 1 ) = 2 √ π η 1 0 e −v 2 dv From this the local Nusselt number is determined as Nu x =− φ (0) √ 2 Re 1/2 x = 2 √ π Pr 2 Re x 1/2 = 0.565Re 1/2 x · Pr 1/2 (6.44) and for Pr 1, the nondimensional stream function near the wall is expressed as f(η) = f(0) 0! + f (0) 1! η + f (0) 2! η 2 + f (0) 3! η 3 +··· (6.45) However, f(0) = f (0) = 0 and the momentum equation shows that f (0) = 0. This results in f(η) = f (0) 2! η 2 = 0.332 √ 2 η 2 (6.46) upon using f (0) = √ 2 f B (0) = 0.332 √ 2 where the subscript B refers to the Blasius variables defined in Section 6.4.2. The energy equation then becomes φ (η) + 0.332Pr √ 2 η 2 φ (η) = 0 (6.47) This can be integrated and evaluated at the surface to yield − φ (0) √ 2 = 0.339Pr 1/3 (6.48a) Nu x = 0.339Re 1/2 x · Pr 1/3 (6.48b) BOOKCOMP, Inc. — John Wiley & Sons / Page 461 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 461 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 [461], (23) Lines: 1016 to 1085 ——— 4.72226pt PgVar ——— Normal Page * PgEnds: Eject [461], (23) 6.4.6 Incompressible Flow Past a Flat Plate with Viscous Dissipation Using the Blasius normalized variables for the flow and then defining the normalized temperature as θ(η) = 2c p (T − T ∞ ) U 2 the energy equation and thermal boundary conditions become θ (η) + 1 2 Pr · f(η)θ (η) + 2Pr · f (η) 2 = 0 (6.49) and θ(η =∞) = 0 and θ(η = 0) = 2c p (T 0 − T ∞ ) U 2 = θ o (constant) (6.50) The solution to eq. (6.49), which is a nonhomogeneous equation, consists of the superposition of a homogeneous part: θ H = C 1 φ(η) − C 2 and a particular solution θ P = θ AW (η) where the subscript AW refers to an adiabatic wall condition. The governing equations and boundary conditions for these are θ AW + 1 2 Pr · f θ AW + 2Pr(f ) 2 = 0 (6.51) φ + 1 2 Pr · f φ = 0 (6.52) and θ AW (0) = θ AW (∞) = 0 and φ(0) = 1 − φ(∞) (6.53) For the boundary conditions for the complete problem to be satisfied, θ(∞) = 0 = θ AW (∞) + C 1 φ(∞) + C 2 which leads to C 1 =−C 2 and θ(0) = 2c p (T 0 − T ∞ ) U 2 = θ AW (0) + C 1 φ(0) + C 2 = 2c p (T AW − T 0 ) U 2 + C 2 or . expressions for the surface heat flux and the Nusselt number are BOOKCOMP, Inc. — John Wiley & Sons / Page 457 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS. C 2 ) 3 m ¯ C 2 ν 2 k 4 x 2m−1 x 4q−3 = β BOOKCOMP, Inc. — John Wiley & Sons / Page 455 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 455 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 [455],. similarity means that BOOKCOMP, Inc. — John Wiley & Sons / Page 453 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER FROM SINGLE OBJECTS IN UNIFORM FLOW 453 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 [453],