Heat Transfer Handbook part 58 pptx

BOOKCOMP, Inc. — John Wiley & Sons / Page 562 / 2nd Proofs / Heat Transfer Handbook / Bejan 562 NATURAL CONVECTION 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 [562], (38) Lines: 978 to 1029 ——— -0.52994pt PgVar ——— Normal Page * PgEnds: Eject [562], (38) Here Nu L,q represents the Nusselt number at x = L and Gr ∗ = gβq  L 4 /kν 2 .Ina later study, Vliet and Ross (1975) obtained a closer corroboration for data in air with the following relationships: Nu x,q =  0.55(Gr ∗ x · Pr) 0.2 for laminar flow (7.79) 0.17(Gr ∗ x · Pr) 0.25 for turbulent flow (7.80) Nu q can be obtained by computing the mean temperature difference and using the overall heat transfer rate provided by Vliet and Ross (1975). 7.7.2 Inclined and Horizontal Flat Surfaces As discussed earlier, the results obtained for vertical surfaces may be employed for surfaces inclined at an angle γ up to about 45° with the vertical, by replacing g with g cos γ in the Grashof number. For inclined surfaces with constant heat flux, Vliet and Ross (1975) have suggested the use of eq. (7.74) for laminar flow, with the replacement of Gr ∗ x by Gr ∗ x cos γ for both upward- and downward-facing heated inclined surfaces. In the turbulent region also, eq. (7.76) is suggested, with Gr ∗ x replaced by Gr ∗ x cos γ for an upward-facing heated surface and with Gr ∗ x replaced by Gr ∗ x cos 2 γ for a downward-facing surface. Several correlations for inclined surfaces, under various thermal conditions, were given by Fujii and Imura (1972). For an inclined plate with heated surface facing upward with approximately constant heat flux, the correlation obtained is of the form Nu q = 0.14[(Gr · Pr) 1/3 − (Gr cr · Pr) 1/3 ] + 0.56(Gr cr · Pr cos γ) 1/4 for 10 5 < Gr · Pr cos γ < 10 11 and 15° < γ < 75° (7.81) where Gr cr is the critical Grashof number at which the Nusselt number starts deviating from the laminar relationship, which is the second expression on the right-hand side of eq. (7.81). This correlation applies for Gr > Gr cr . The value of Gr cr is also given for various inclination angles. For γ = 15, 30, 60, and 70°, Gr cr is given as 5 × 10 9 , 2 × 10 9 ,10 8 , and 10 6 , respectively. For inclined heated surfaces facing downward, the expression given is Nu q = 0.56(Gr · Pr cos γ) 1/4 for 10 5 < Gr · Pr cos γ < 10 11 , γ < 88° (7.82) The fluid properties are evaluated at T w −0.25(T w −T ∞ ), and β at T ∞ +0.25(T w −T ∞ ). For horizontal surfaces, several classical expressions exist. For heated isothermal surfaces facing downward (lower surface of heated plate), or cooled ones facing upward (upper surface of cooled plate), the correlation given by McAdams (1954) which has been used extensively, is Nu = 0.27Ra 1/4 for 10 5  Ra  10 10 (7.83) Fujii and Imura (1972) give the corresponding correlation as BOOKCOMP, Inc. — John Wiley & Sons / Page 563 / 2nd Proofs / Heat Transfer Handbook / Bejan EMPIRICAL CORRELATIONS 563 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 [563], (39) Lines: 1029 to 1081 ——— 5.2721pt PgVar ——— Normal Page * PgEnds: Eject [563], (39) Nu = 0.58Ra 1/5 for 10 6 ≤ Ra ≤ 10 11 (7.84) Over the overlapping range of the two studies by Fujii and Imura (1972) and Rotem and Claassen (1969), the agreement between the two Nusselt numbers is fairly good. For the heated isothermal horizontal surface facing upward and cold surface facing downward, the correlations for heat transfer are given by McAdams (1954) as Nu =  0.54Ra 1/4 for 10 4  Ra  10 7 (7.85) 0.15Ra 1/3 for 10 7  Ra  10 11 (7.86) The corresponding correlation given by Fujii and Imura (1972) for an approximately uniform heat flux condition is Nu q = 0.14Ra 1/3 for Ra > 2 × 10 8 (7.87) 7.7.3 Cylinders and Spheres A considerable amount of information exists on natural convection heat transfer from cylinders. For vertical cylinders of large diameter, ascertained from eq. (7.47), the correlations for vertical flat surfaces may be employed. For cylinders of small diameter, correlations for Nu are suggested in terms of the Rayleigh number Ra, where Ra and Nu are based on the diameter D of the cylinder. The horizontal cylinder has been of interest to several investigators. McAdams (1954) gave correlation for isothermal cylinders as Nu =  0.53Ra 1/4 for 10 4 < Ra < 10 9 (7.88) 0.13Ra 1/3 for 10 9 < Ra < 10 12 (7.89) For smaller values of Ra, graphs are presented by McAdams (1954). A general expression of the form Nu = C · Ra n is given by Morgan (1975), with C and n presented in tabular form. Churchill and Chu (1975b) have given a correlation covering a wide range of Ra, Ra ≤ 10 12 , for isothermal cylinders as Nu =  0.60 + 0.387  Ra [1 + (0.559/Pr) 9/16 ] 16/9  1/6  2 (7.90) This correlation is recommended for horizontal cylinders since it is convenient to use and agrees closely with experimental results. For natural convection from spheres, too, several experimental studies have pro- vided heat transfer correlations. Amato and Tien (1972) have listed the correlations for Nu obtained from various investigations of heat and mass transfer. In a review pa- per, Yuge (1960) suggested the following correlation for heat transfer from isothermal spheres in air and gases over a Grashof number range 1 < Gr < 10 5 , where Gr and Nu are based on the diameter D: BOOKCOMP, Inc. — John Wiley & Sons / Page 564 / 2nd Proofs / Heat Transfer Handbook / Bejan 564 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 [564], (40) Lines: 1081 to 1109 ——— * 528.0pt PgVar ——— Normal Page * PgEnds: PageBreak [564], (40) TABLE 7.2 Summary of Natural Convection Correlations for External Flows over Isothermal Surfaces Geometry Recommended Correlation Range Reference Vertical flat surfaces Nu =  0.825 + 0.387Ra 1/6 [1 + (0.492/Pr) 9/16 ] 8/27  2 10 −1 < Ra < 10 12 Churchill and Chu (1975a) Inclined flat surfaces Above equation with g replaced by g cos γγ≤ 60° Horizontal flat surfaces Nu = 0.54Ra 1/4 10 5 ≤ Ra ≤ 10 7 Heated, facing upward Nu = 0.15Ra 1/3 10 7 ≤ Ra ≤ 10 10 McAdams (1954) Heated, facing downward Nu = 0.27Ra 1/4 3 × 10 5 ≤ Ra ≤ 3 × 10 10 McAdams (1954) Horizontal cylinders Nu =  0.60 + 0.387  Ra [1 + (0.559/Pr) 9/16 ] 16/9  1/6  2 10 −5 ≤ Ra ≤ 10 12 Churchill and Chu (1975b) Spheres Nu = 2 + 0.43Ra 1/4 Pr = 1 and 1 < Ra < 10 5 Yuge (1960) BOOKCOMP, Inc. — John Wiley & Sons / Page 565 / 2nd Proofs / Heat Transfer Handbook / Bejan EMPIRICAL CORRELATIONS 565 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 [565], (41) Lines: 1109 to 1162 ——— -0.3219pt PgVar ——— Normal Page * PgEnds: Eject [565], (41) Nu = 2 + 0.43Ra 1/4 for Pr = 1 and 1 < Ra < 10 5 (7.91) For heat transfer in water, Amato and Tien (1972) obtained the correlation for isother- mal spheres as Nu = 2 + C ·Ra 1/4 for 3 × 10 5 ≤ Ra ≤ 8 × 10 8 (7.92) with C = 0.500 ± 0.009, which gave a mean deviation of less than 11%. A general correlation applicable for Pr ≥ 0.7 and Ra  10 11 is given by Churchill (1983) as Nu = 2 + 0.589Ra 1/4  1 + (0.469/Pr) 9/16  4/9 (7.93) Several of the important correlations presented earlier are summarized in Table 7.2. Correlations for various other geometries are given by Churchill (1983) and Raithby and Hollands (1985). Nu and Ra are based on the height L for a vertical plate, length L for inclined and horizontal surfaces, and diameter D for horizontal cylinders and spheres. All fluid properties are evaluated at the film temperature T f = (T w +T ∞ )/2. 7.7.4 Enclosures As mentioned earlier, the heat transfer across a vertical rectangular cavity is largely by conduction for Ra  10 3 , which implies a Nusselt number Nu of 1.0. For larger Ra, Catton (1978) has given the following correlation for the aspect ratio H/d in the range 2 to 10 and Pr < 10 5 : Nu = 0.22  Pr 0.2 + Pr Ra  0.28  H d  −1/4 (7.94) where the Nusselt and Rayleigh numbers are based on the distance d between the vertical walls and the temperature difference between them. For an aspect ratio be- tween 1 and 2, the coefficient in this expression was changed from 0.22 to 0.18 and the exponent from 0.28 to 0.29, with the aspect ratio dependence dropped. Similarly, correlations are given for higher aspect ratios in the literature. For horizontal cavities heated from below, the Nusselt number Nu is 1.0 for Rayleigh number Ra  1708, as discussed earlier. Globe and Dropkin (1959) gave the following correlation for such cavities at larger Ra, 3 × 10 5 < Ra < 7 × 10 9 : Nu = 0.069Ra 1/3 · Pr 0.074 (7.95) For inclined cavities, Hollands et al. (1976) gave the following correlation for air as the fluid with H/d  12 and γ < γ ∗ on the basis of several experimental studies: Nu = 1 + 1.44  1 − 1708 Ra cos γ  1 − 1708(sin 1.8γ) 1.6 Ra cos γ  +   Ra cos γ 5830  1/3 − 1  (7.96) BOOKCOMP, Inc. — John Wiley & Sons / Page 566 / 2nd Proofs / Heat Transfer Handbook / Bejan 566 NATURAL CONVECTION 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 [566], (42) Lines: 1162 to 1230 ——— 1.2814pt PgVar ——— Normal Page PgEnds: T E X [566], (42) where γ is the inclination with the horizontal, γ ∗ is a critical angle tabulated by Hollands et al. (1976), and the term in the square brackets is set equal to zero if the quantity within these brackets is negative. This equation uses the stability limit of Ra = 1708 for a horizontal layer, given earlier. For a horizontal enclosure heated from below, with air as the fluid, Hollands et al. (1975) gave the correlation Nu = 1 + 1.44  1 − 1708 Ra  +   Ra 5830  1/3 − 1  (7.97) Similarly, correlations for other fluids, geometries, and thermal conditions are given in the literature. 7.8 SUMMARY In this chapter we discuss the basic considerations relevant to natural convection flows. External and internal buoyancy-induced flows are considered, and the govern- ing equations are obtained. The approximations generally employed in the analysis of these flows are outlined. The important dimensionless parameters are derived in order to discuss the importance of the basic processes that govern these flows. Lami- nar flows for various surfaces and thermal conditions are discussed, and the solutions obtained are presented, particularly those derived from similarity analysis. The heat transfer results and the characteristics of the resulting velocity and temperature fields are discussed. Also considered are transient and turbulent flows. The governing equa- tions for turbulent flow are given, and experimental results for various flow configura- tions are presented. The frequently employed empirical correlations for heat transfer by natural convection from various surfaces and enclosures are also included. Thus, this chapter presents the basic aspects that underlie natural convection and the heat transfer correlations that may be employed for practical applications. NOMENCLATURE Roman Letter Symbols c p specific heat at constant pressure, J/kg · K D diameter of cylinder or sphere, m f stream function, dimensionless F body force per unit volume, N/m 3 g gravitational acceleration, m/s 2 Gr Grashof number, dimensionless Gr x local Grashof number, dimensionless Gr * heat flux Grashof number, dimensionless h x local heat transfer coefficient, W/m 2 · K ¯ h average heat transfer coefficient, W/m 2 · K h φ heat transfer coefficient at angular position φ, W/m 2 · K BOOKCOMP, Inc. — John Wiley & Sons / Page 567 / 2nd Proofs / Heat Transfer Handbook / Bejan NOMENCLATURE 567 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 [567], (43) Lines: 1230 to 1249 ——— 1.82222pt PgVar ——— Normal Page PgEnds: T E X [567], (43) k thermal conductivity, W/m · K L characteristic length, height of vertical plate, m m, n exponents in exponential and power law distributions, dimensionless M,N constants employed for exponential and power law distributions of surface temperature, dimensionless Nu x local Nusselt number, dimensionless [ = h x x/k ] Nu average Nusselt number for an isothermal surface, dimensionless Nu q average Nusselt number for a uniform heat flux surface, dimensionless Nu φ local Nusselt number at angular position φ, dimensionless p pressure, Pa Pr Prandtl number, dimensionless q total heat transfer, W q  x local heat flux, W/m 2 q  constant surface heat flux, W/m 2 q  volumetric heat source, W/m 3 Ra Rayleigh number, dimensionless [ = Gr · Pr ] Ra x local Rayleigh number, dimensionless [ = Gr x · Pr ] Sr Strouhal number, dimensionless t time, s t c characteristic time, s ∆T temperature difference, K [= T w − T ∞ ] T local temperature, K T w wall temperature, K plume centerline temperature, K T ∞ ambient temperature, K u, v, w velocity components in x,y, and z directions, respectively, m/s V velocity vector, m/s V c convection velocity, m/s x,y, z coordinate distances, m Greek Letter Symbols α thermal diffusivity, m 2 /s β coefficient of thermal expansion, K −1 γ inclination with the vertical, degrees or radians δ velocity boundary layer thickness, m δ T thermal boundary layer thickness, m ε H eddy diffusivity, m 2 /s ε M eddy viscosity, m 2 /s η similarity variable, dimensionless θ temperature, dimensionless [ = (T − T ∞ )/(T w − T ∞ ) ] µ dynamic viscosity, Pa · s BOOKCOMP, Inc. — John Wiley & Sons / Page 568 / 2nd Proofs / Heat Transfer Handbook / Bejan 568 NATURAL CONVECTION 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 [568], (44) Lines: 1249 to 1297 ——— 4.98828pt PgVar ——— Custom Page (2.0pt) PgEnds: T E X [568], (44) ν kinematic viscosity, m 2 /s Φ v viscous dissipation, s −2 ψ stream function, m 2 /s REFERENCES Abib, A., and Jaluria, Y. 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Heat Transfer, 82, 214–220. . number, dimensionless Gr * heat flux Grashof number, dimensionless h x local heat transfer coefficient, W/m 2 · K ¯ h average heat transfer coefficient, W/m 2 · K h φ heat transfer coefficient at angular. Flow in a Partially Open Enclosure, Numer. Heat Transfer, 14, 235–254. Abib, A., and Jaluria, Y. (1995). Turbulent Penetrative and Recirculating Flow in a Compart- ment Fire, J. Heat Transfer, . and Tien, C. (1972). Free Convection Heat Transfer from Isothermal Spheres in Water, Int. J. Heat Mass Transfer, 15, 327–339. Batchelor, G. K. (1954). Heat Transfer by Free Convection across a