Heat Transfer Handbook part 42 ppt

BOOKCOMP, Inc. — John Wiley & Sons / Page 402 / 2nd Proofs / Heat Transfer Handbook / Bejan 402 FORCED CONVECTION: INTERNAL 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 [402], (8) Lines: 331 to 362 ——— 2.89604pt PgVar ——— Normal Page * PgEnds: Eject [402], (8) TABLE 5.1 Scale Drawing of Five Different Ducts That Have the Same Hydraulic Diameter Cross Section Diagram Circular Square Equilateral triangle Rectangular (4:1) Infinite parallel plates Source: Bejan (1995). Table 5.1 shows five duct cross sections that have the same hydraulic diameter. The hydraulic diameter of the round tube coincides with the tube diameter. The hydraulic diameter of the channel formed between two parallel plates is twice the spacing between the plates. For cross sections shaped as regular polygons, D h is the diameter of the circle inscribed inside the polygon. In the case of highly asymmetric cross sections, D h scales with the smaller of the two dimensions of the cross section. The general pressure drop relationship (5.2) is most often written in terms of hydraulic diameter, BOOKCOMP, Inc. — John Wiley & Sons / Page 403 / 2nd Proofs / Heat Transfer Handbook / Bejan LAMINAR FLOW AND PRESSURE DROP 403 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 [403], (9) Lines: 362 to 436 ——— * 18.75684pt PgVar ——— Normal Page PgEnds: T E X [403], (9) ∆P = f 4L D h  1 2 ρU 2  (5.16) To calculate ∆P , the friction factor f must be known and it can be derived from the flow solution. The friction factors derived from the Hagen–Poiseuille flows described by eqs. (5.7) and (5.9) are f =      24 Re D h D h = 2D parallel plates (D = spacing) (5.17) 16 Re D h D h = D round tube (D = diameter) (5.18) Equations (5.17) and (5.18) hold for laminar flow (Re D h ≤ 2000). Friction factors for other cross-sectional shapes are reported in Tables 5.2 and 5.3. Additional results can be found in Shah and London (1978). All Hagen–Poiseuilleflows are characterized by TABLE 5.2 Effect of Cross-Sectional Shape on f and Nu in Fully Developed Duct Flow Nu = hD h /k Cross Section f/Re D h B = πD 2 h /4 A Uniform q  Uniform T 0 13.3 0.605 3 2.35 14.2 0.785 3.63 2.89 16 1 4.364 3.66 18.3 1.26 5.35 4.65 24 1.57 8.235 7.54 24 1.57 5.385 4.86 Source: Bejan (1995). BOOKCOMP, Inc. — John Wiley & Sons / Page 404 / 2nd Proofs / Heat Transfer Handbook / Bejan 404 FORCED CONVECTION: INTERNAL 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 [404], (10) Lines: 436 to 477 ——— 0.39122pt PgVar ——— Normal Page PgEnds: T E X [404], (10) TABLE 5.3 Friction Factors and Nusselt Numbers for Heat Transfer to Laminar Flow through Ducts with Regular Polygonal Cross Sections Nu = hD h /k f · Re D h Uniform Heat Flux Isothermal Wall Fully Fully Fully Developed Developed Slug Developed Slug Cross Section Flow Flow Flow Flow Flow Square 14.167 3.614 7.083 2.980 4.926 Hexagon 15.065 4.021 7.533 3.353 5.380 Octagon 15.381 4.207 7.690 3.467 5.526 Circle 16 4.364 7.962 3.66 5.769 Source: Data from Asako et al. (1988). f = C Re D h (5.19) where the constant C depends on the shape of the duct cross section. It was shown in Bejan (1995) that the duct shape is represented by the dimensionless group B = πD h /4 A duct (5.20) and that f · Re D h (or C) increases almost proportionally with B. This correlation is illustrated in Fig. 5.4 for the duct shapes documented in Table 5.2. 5.3 HEAT TRANSFER IN FULLY DEVELOPED FLOW 5.3.1 Mean Temperature Consider the stream shown in Fig. 5.5, and assume that the duct cross section A is not specified. According to the thermodynamics of open systems, the first law for the control volume of length dx is q  p =˙m dh/dx, where h is the bulk enthalpy of the stream. When the fluid is an ideal gas (dh = c p dT m ) or an incompressible liquid with negligible pressure changes (dh = cdT m ), the first law becomes dT m dx = q  p ˙mc p (5.21) In heat transfer, the bulk temperature T m is known as mean temperature. It is related to the bundle of ministreams of enthalpy (ρuc p TdA) that make up the bulk enthalpy stream (h) shown in the upper left corner of Fig. 5.5. From this observation it follows that the definition of T m involves a u-weighted average of the temperature distribution over the cross section, BOOKCOMP, Inc. — John Wiley & Sons / Page 405 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER IN FULLY DEVELOPED FLOW 405 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 [405], (11) Lines: 477 to 484 ——— 0.23903pt PgVar ——— Normal Page PgEnds: T E X [405], (11) Figure 5.4 Effect of the cross-sectional shape (the number B) on fully developed friction and heat transfer in a straight duct. (From Bejan, 1995.) T m = 1 UA  A uT dA (5.22) In internal convection, the heat transfer coefficient h = q  /∆T is based on the difference between the wall temperature (T 0 ) and the mean temperature of the stream: namely, h = q  /(T 0 − T m ). 5.3.2 Thermally Fully Developed Flow By analogy with the developing velocity profile described in connection with Fig. 5.1, there is a thermal entrance region of length X T . In this region the thermal boundary BOOKCOMP, Inc. — John Wiley & Sons / Page 406 / 2nd Proofs / Heat Transfer Handbook / Bejan 406 FORCED CONVECTION: INTERNAL 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 [406], (12) Lines: 484 to 500 ——— 0.05212pt PgVar ——— Normal Page PgEnds: T E X [406], (12) ␳udA m . qЉ qЉ r ␪ v u x x x+dx TdT mm ϩ T m Figure 5.5 Nomenclature for energy conservation in a duct segment. (From Bejan, 1995.) layers grow and effect changes in the distribution of temperature over the duct cross section. Estimates for X T are given in Section 5.4.1. Downstream from x ∼ X T the thermal boundary layers have merged and the shape of the temperature profile across the duct no longer varies. For a round tube of radius r 0 , this definition of a fully developed temperature profile is T 0 (x) −T(r,x) T 0 (x) −T m (x) = φ  r r 0  (5.23) The function φ(r/r 0 ) represents the r-dependent shape (profile) that does not depend on the downstream position x. The alternative to the definition in eq. (5.23) is the scale analysis of the same regime (Bejan, 1995), which shows that the heat transfer coefficient must be con- stant (x-independent) and of order k/D. The dimensionless version of this second definition is the statement that the Nusselt number is a constant of order 1: Nu = hD k = D ∂T /∂r   r=r 0 T 0 − T m = O(1) (5.24) The second part of the definition refers to a tube of radius r 0 . The Nu values compiled in Tables 5.2 and 5.3 confirm the constancy and order of magnitude associated with thermally fully developed flow. The Nu values also exhibit the approximate propor- tionality with the B number that characterizes the shape of the cross section (Fig. 5.4). In Table 5.3, slug flow means that the velocity is distributed uniformly over the BOOKCOMP, Inc. — John Wiley & Sons / Page 407 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER IN DEVELOPING FLOW 407 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 [407], (13) Lines: 500 to 535 ——— 7.09703pt PgVar ——— Normal Page PgEnds: T E X [407], (13) cross section, u = U . Noteworthy are the Nu values for a round tube with uniform wall heat flux Nu = 48 11 = 4.364 uniform wall heat flux and a round tube with isothermal wall Nu = 3.66 isothermal wall 5.4 HEAT TRANSFER IN DEVELOPING FLOW 5.4.1 Thermal Entrance Region The heat transfer results listed in Tables 5.2 and 5.3 apply to laminar flow regions where the velocity and temperature profiles are fully developed. They are valid in the downstream section x, where x>max(X, X T ). The flow development length X is given by eq. (5.2). The thermal length X T is determined from a similar scale analysis by estimating the distance where the thermal boundary layers merge, as shown in Fig. 5.6. When Pr  1, the thermal boundary layers are thicker than the velocity boundary layers, and consequently, X T  X. The Prandtl number Pr is the ratio of the molecular momentum and thermal diffusivities, ν/α. When Pr  1, the thermal boundary layers are thinner, and X T is considerably greater than X. The scale analysis of this problem shows that for both Pr  1 and Pr  1, the relationship between X T and X is (Bejan, 1995) X T X ≈ Pr (5.25) Pr 1 Pr 1 0 0 X X XX T ϳ Pr XX T ϳ Pr x Figure 5.6 Prandtl number effect on the flow entrance length X and the thermal entrance length X T . (From Bejan, 1995.) BOOKCOMP, Inc. — John Wiley & Sons / Page 408 / 2nd Proofs / Heat Transfer Handbook / Bejan 408 FORCED CONVECTION: INTERNAL 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 [408], (14) Lines: 535 to 553 ——— 1.927pt PgVar ——— Normal Page * PgEnds: Eject [408], (14) Equations (5.25) and (5.2) yield the scale X T ≈ 10 −2 Pr · D h · Re D h (5.26) which is valid over the entire Pr range. 5.4.2 Thermally Developing Hagen–Poiseuille Flow When Pr  1, there is a significant length of the duct (X<x<X T ) over which the velocity profile is fully developed, whereas the temperature profile is not. If the x-independent velocity profile of Hagen–Poiseuille flow is assumed, it is possible to solve the energy conservation equation and determine, as an infinite series, the temperature field (Graetz, 1883). The Pr =∞curve in Fig. 5.7 shows the main features of the Graetz solution for heat transfer in the entrance region of a round tube with an isothermal wall (T 0 ). The Reynolds number Re D = UD/ν is based on the tube diameter D and the mean velocity U. The bulk dimensionless temperature of the stream (θ ∗ m ), the local Nusselt number (Nu x ), and the averaged Nusselt number (Nu 0−x ) are defined by 100 10 3 Nu x Pr = ϱ 5 2 0.7 0.01 0.1 1 xD/ Re Pr D ( ( 1/2 3.66 0.1 1 ␪ *m Nu (Pr = ) 0Ϫx ϱ ␪ϱ * (Pr = ) m Figure 5.7 Heat transfer in the entrance region of a round tube with isothermal wall. (From Bejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.) BOOKCOMP, Inc. — John Wiley & Sons / Page 409 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER IN DEVELOPING FLOW 409 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 [409], (15) Lines: 553 to 618 ——— 1.28441pt PgVar ——— Normal Page * PgEnds: Eject [409], (15) θ ∗ m = T m − T 0 T in − T 0 (5.27) Nu x = q  x D k(T 0 − T m ) (5.28) Nu 0−x = q  0−x D k ∆T m (5.29) In these expressions, T m (x) is the local mean temperature, T in the stream inlet tem- perature, q  x the local wall heat flux, and q  0−x the heat flux averaged from x = 0to x, and ∆T lm the logarithmic mean temperature difference, ∆T lm = [ T 0 − T in (x) ] − (T 0 − T m ) ln [ T 0 − T m (x)/(T 0 − T in ) ] (5.30) The dimensionless longitudinal position plotted on the abscissa is also known as x ∗ : x ∗ = x/D Re D · Pr (5.31) This group, and the fact that the knees of the Nu curves occur at x 1/2 ∗ ≈ 10 −1 , support the X T estimate anticipated by eq. (5.26). The following analytical expressions are recommended by a simplified alternative to Graetz’s series solution (L ´ ev ˆ eque, 1928; Drew, 1931). The relationships for the Pr =∞curves shown in Fig. 5.7 are (Shah and London, 1978) Nu x =  1.077x −1/3 ∗ − 0.70 x ∗ ≤ 0.01 3.657 + 6.874(10 3 x ∗ ) −0.488 e −57.2x ∗ x ∗ > 0.01 (5.32) Nu 0−x =      1.615x −1/3 ∗ − 0.70 x ∗ ≤ 0.005 1.615x −1/3 ∗ − 0.20 0.005 <x ∗ < 0.03 3.657 + 0.0499/x ∗ x ∗ > 0.03 (5.33) The thermally developing Hagen–Poiseuille flow in a round tube with uniform heat flux q  can be analyzed by applying Graetz’s method (the Pr =∞curves in Fig. 5.8). The results for the local and overall Nusselt numbers are represented within 3% by the equations (see also Shah and London, 1978) Nu x ∗ =      3.302x −1/3 ∗ − 1.00 x ∗ ≤ 0.00005 1.302x −1/3 ∗ − 0.50 0.00005 <x ∗ ≤ 0.0015 4.364 + 8.68(10 3 x ∗ ) −0.506 e −41x ∗ x ∗ > 0.001 (5.34) Nu 0−x =  1.953x −1/3 ∗ x ∗ ≤ 0.03 4.364 + 0.0722/x ∗ x ∗ > 0.03 (5.35) BOOKCOMP, Inc. — John Wiley & Sons / Page 410 / 2nd Proofs / Heat Transfer Handbook / Bejan 410 FORCED CONVECTION: INTERNAL 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 [410], (16) Lines: 618 to 654 ——— 2.2831pt PgVar ——— Normal Page PgEnds: T E X [410], (16) 100 10 3 Nu x Pr = ϱ 5 2 0.7 0.01 0.1 1 xD/ Re Pr D ( ( 1/2 4.36 Nu (Pr = ) 0Ϫx ϱ Figure 5.8 Heat transfer in the entrance region of a round tube with uniform heat flux. (From Bejan, 1995; drawn based on data from Shah and London, 1978, and Hornbeck, 1965.) where Nu x ∗ = q  D k [ T 0 (x) −T m (x) ] Nu 0−x = q  D k ∆T avg with ∆T avg =  1 x  x 0 dx T 0 (x) −T m (x)  −1 (5.36) Analogous results are available for the heat transfer to thermally developing Hagen–Poiseuille flow in ducts with other cross-sectional shapes. The Nusselt num- bers for a parallel-plate channel are shown in Fig. 5.9. The curves for a channel with isothermal surfaces are approximated by (Shah and London, 1978) Nu 0−x =  1.233x −1/3 ∗ + 0.40 x ∗ ≤ 0.001 7.541 + 6.874(10 3 x ∗ ) −0.488 e −245x ∗ x ∗ > 0.001 (5.37) BOOKCOMP, Inc. — John Wiley & Sons / Page 411 / 2nd Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER IN DEVELOPING FLOW 411 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 [411], (17) Lines: 654 to 688 ——— * 16.42421pt PgVar ——— Normal Page PgEnds: T E X [411], (17) Nu 0−x =      1.849x −1/3 ∗ x ∗ ≤ 0.0005 1.849x −1/3 ∗ + 0.60 0.0005 <x ∗ ≤ 0.006 7.541 + 0.0235/x ∗ x ∗ > 0.006 (5.38) If the plate-to-plate spacing is D, the Nusselt numbers are defined as Nu x = q  (x)D h k [ T 0 − T m (x) ] Nu 0−x = q  0−x D h k ∆T lm where D h = 2D and ∆T lm is given by eq. (5.30). The thermal entrance region of the parallel-plate channel with uniform heat flux and Hagen–Poiseuille flow is characterized by (Shah and London, 1978) Nu x =    1.490x −1/3 ∗ x ∗ ≤ 0.0002 1.490x −1/3 ∗ − 0.40 0.0002 <x ∗ ≤ 0.001 8.235 + 8.68(10 3 x ∗ ) 0.506 e −164x ∗ x ∗ > 0.001 (5.39) 100 10 3 Nu , Nu xx0Ϫ Nu x Nu x Nu 0Ϫx Nu 0Ϫx 0.01 0.1 1 xD/ Re Pr h D h ( ( 1/2 8.23 7.54 Uniform wall temperature Uniform wall heat flux Figure 5.9 Heat transfer in the thermal entrance region of a parallel-plate channel with Hagen–Poiseuille flow. (From Bejan,1995; drawn based on data from Shah and London, 1978.) . / Heat Transfer Handbook / Bejan HEAT TRANSFER IN FULLY DEVELOPED FLOW 405 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 [405],. Proofs / Heat Transfer Handbook / Bejan HEAT TRANSFER IN DEVELOPING FLOW 407 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 [407],. heat flux Nu = 48 11 = 4.364 uniform wall heat flux and a round tube with isothermal wall Nu = 3.66 isothermal wall 5.4 HEAT TRANSFER IN DEVELOPING FLOW 5.4.1 Thermal Entrance Region The heat transfer