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where u 0 is the maximum velocity (in the axis). The problem then reduces to ®nding u 0 . It is obvious that uyY z thus obtained does not identically match the true solution to the problem. The maximum value of the velocity results by imposing that (3.152) veri®es (3.151) integrated (averaged) over the duct cross section, namely, ab dP dx m a 2 À a 2 b 2 À b 2 d 2 u dy 2 d 2 u dz 2 dzdyX 3X153 Consequently, ab dP dx À 16 3 mu 0 b a a b Y 3X154 and using the de®nition of the average, U def 1 ab a 2 À a 2 b 2 À b 2 udz dyY 3X155 yields u 0 9 4 U X 3X156 Finally, this result produces the following expression for the friction factor: f a 2 b 2 a b 2 24 Re D h Y where D h 2ab a b X 3X157 3.2.4 HEAT TRANSFER IN THE FULLY DEVELOPED REGION The key problem of heat transfer in duct ¯ows is to determine the relation- ship between the wall-to-stream temperature drop and its associated heat transfer rate. For ¯ow in a circular duct of radius r 0 and with average longitudinal velocity U , the mass ¯ow rate is m rpr 2 0 U (Fig. 3.20, [10]). The heat transfer rate from the wall to the stream should equal the change in the enthalpy of the stream. To verify this we write the ®rst principle for a control volume of length dx, q HH 2pr 0 dx mh xdx À h x X 3X158 In the perfect gas limit of the ¯uid dh c P dT m , or in the limit of an incompressible ¯ow and under a negligible pressure variation dh c P dT m , this balance equation implies dT m dx 2 r 0 q HH rcU Y 3X159 where, for incompressible ¯ows, c P was replaced by c. The temperature T m that appears in the energy balance written for the control volume is the average temperature of the stream, and it indicates that the actual tempera- 528 Principles of Heat Transfer Heat Transfer ture ®eld in the stream is not uniform. Its de®nition results from the ®rst principle applied to a stream tube, that is, q HH 2pr 0 dx d A ruc P TdA,as T m def 1 rc P U 1 A A ruc P TdAX 3X160 For ducts with uniform-temperature walls, this means T m 1 U 1 pr 2 0 A uTdAX 3X161 As the temperature varies in every cross section of the duct, there exists a wall-to-stream temperature drop DT T w À T m . The heat transfer coef®cient is then h def q HH T w À T m k dT dr r r 0 DT Y 3X162 where q HH is positive if the heat transfer is wall 6stream. 3.2.5 THE FULLY DEVELOPED TEMPERATURE PROFILE The heat transfer coef®cient may be evaluated provided the temperature ®eld T x Y y, and hence the energy boundary value problem for the speci®ed boundary conditions, is solved ®rst. For example, for the stationary laminar ¯ow in a straight circular duct, the energy equation is 1 a u dT dx v dT dy d 2 T dr 2 1 r dT dr d 2 T dx 2 X 3X163 In the hydrodynamic fully developed region, v 0 and u ur , which implies ur a dT dx d 2 T dr 2 1 r dT dr d 2 T dx 2 X 3X164 Figure 3.20 Heat transfer in the fully developed region for ¯ow in a cylindrical duct. 3. Convection Heat Transfer 529 Heat Transfer This equation indicates the balance Axial convection $ Radial conductionY Axial conduction with the following scales: U a q HH Drc P U z}|{ Convection $ DT D 2 radial Y 1 x q HH Drc P U X Longitudinal z}|{ Conduction 3X165 Note that en route to this scaling relation we used the relation dT dx $ q HH Drc P U Y 3X166 introduced by (3.159). Apparently, the radial conduction is a central term in (3.165) Ð without its contribution there is no heat transfer associated to the internal ¯ow. The scales (3.165) multiplied by D 2 aDT and the de®nition of the heat transfer coef®cient, h q HH aDT , yield hD k z}|{ Convection $ 1 Radial Y hD k 2 a DU 2 X Longitudinal z}|{ Conduction 3X167 The bottom line to this analysis is that for large PeÂclet numbers, Pe D , the axial conduction heat transfer may be negligibly small, and the energy equation becomes ur a dT dx d 2 T dr 2 1 r dT dr Pe D UD a ) 1 X 3X168 Furthermore, from the balance Axial convection $ Radial conductionY it follows that the Nusselt number is a constant of order 1, Nu D hD k $ 1Pe D UD a ) 1 X 3X169 The temperature pro®le produced by this analysis corresponds to fully developed ¯ow. It represents the temperature distribution downstream from the two entrance regions X Y X T Y where both u and T are developing. At this point it is important to note that, in the literature, the fully developed temperature pro®le is de®ned through T w À T T w À T m f r r 0 Y 3X170 where T , T w , and T m may be functions of x. This analytic form for T xY r is a consequence of Nu D $ 1, so, Nu D hD k q HH T w À T m D k Y or Nu D D dT dr r r 0 T w À T m $ 1X 3X171 530 Principles of Heat Transfer Heat Transfer Consequently, the variation of dT adr j r r 0 with respect to x is identical to that of T w xÀT m x, and since dT adr is a function of x and r , it follows that Nu D dT adr ar 0 T w xÀT m x f 1 r r 0 O1Y 3X172 which, further integrated with respect to r ar 0 , yields TxY r r 0 T 0 À T m f 2 r r 0 f 3 xX 3X173 Here f 1 , f 2 , and f 3 are functions of r ar 0 and x. 3.2.6 DUCTS WITH UNIFORM HEAT FLUX WALLS When q HH is not a function of x, the ordinary differential equation (3.168) admits an analytical solution, T xY r T w xÀ q HH h f r r 0 Y hence dT dx dT w dx Y or dT w dx dT m dx X 3X174 By virtue of (3.159), dT dx 2 r 0 q HH rc P U const 3X175 Consequently, the temperature at a particular location is a linear function of x, and its slope is proportional to q HH (Fig. 3.21), after [10]. On the other hand, the r -variation of T , respectively fr ar 0 , may be found by solving the energy equation for the thermally fully developed ¯ow. Using (3.168), the temperature pro®le (3.174) and the Hagen±Poiseuille velocity pro®le (3.144) yield À2 hD k 1 À r Ã2 d 2 f dr Ã2 1 r * df dr * Y r * r r 0 Y 3X176 Figure 3.21 Heat transfer in the fully developed region for a cylindrical duct with uniform heat ¯ux walls. 3. Convection Heat Transfer 531 Heat Transfer with Nu D hDak now emerging explicitly. Integrating this equation twice and using the boundary condition f H j r *0 finite results in fr *Y Nu D C 2 À 2Nu D r * 2 2 À r Ã2 4 2 45 X 3X177 The integration constant C 2 may be found by using (3.174), (3.177), and the condition T j r *1 T w , namely, T T w ÀT w À T m Nu 3 8 À r Ã2 2 r Ã4 8 X 3X178 The average temperature drop T w À T m (3.160) is then T w À T m 1 pr 2 0 U 2p 0 r 0 0 T w À T urdrd y 4 1 0 T w À T 1 À r Ã2 r *dr *X 3X179 Consequently, 1 4Nu D 1 0 3 8 À r Ã2 2 r Ã4 8 1 À r Ã2 r *dr * 11 48 Nu D Y 3X180 which means that the Nusselt number for the thermally fully developed Hagen±Poiseuille region is Nu D 48 11 X 3X181 This result is in good agreement with the scale analysis (3.169). Table 3.3 gives the Nusselt number, Nu D hD h ak, for different types of ducts, obtained by integrating the energy equation u a dT dx H 2 T Y 3X182 where the longitudinal temperature gradient, dT adx , may be explicitly obtained via a balance equation of type (3.159), that is, dT m dx q H rc P AU const 3X183 q H is the heat transfer rate per duct unit length, an x-independent quantity. Generally, for the noncircular ducts, the wall temperature on a circumfer- ential outline, T w , is assumed constant and, subsequently, the heat ¯ux is nonuniform in the x-direction. 3.2.7 DUCTS WITH ISOTHERMAL WALLS Figure 3.22, after [10], shows a qualitative sketch of the temperature pro®le for a duct with an isothermal, T w , wall. If the average temperature of the ¯ow at coordinate x 1 in the fully developed region is T 1 , then the heat transfer is 532 Principles of Heat Transfer Heat Transfer driven by the temperature difference T w À T 1 where the ¯ow temperature is increasing downstream, monotonously. Consequently, the heat ¯ux q HH xhT w À T m x 3X184 exhibits the same trend. On the other hand, by scale analysis (3.169), h was shown to be constant. Now, eliminating q HH between (3.184) and (3.159) and then integrating it yields T 0 À T m x T 0 À T 1 e ÀaxÀx 1 ar 2 0 U Nu D Y 3X185 which clearly indicates an exponential decrease with respect to x.TheNu D number that appears in (3.185) may be computed by solving the energy equation (3.168), where the temperature gradient dT adx is written as dT dx d dx T 0 À fT 0 À T m f dT m dx X Merging this result, the Hagen±Poiseuille solution, and the temperature pro®le T T 0 À fT 0 À T m into the energy equation 3X164 leads to the nondimensional form À2Nu D 1 À r Ã2 f d 2 f dr Ã2 1 r à df dr * Y 3X186 which, if we observe that the sign of fr* is reversed, is similar to (3.176). The corresponding boundary conditions may be Axial symmetryX df dr * r à 0 0Y Isothermal wallX fj r à 1 0X 3X187 Figure 3.22 Heat transfer in the fully developed region for a cylindrical duct with isothermal walls. 3. Convection Heat Transfer 533 Heat Transfer If we apply the heat transfer coef®cient de®nition (3.162), the Nusselt number then results as Nu D À2 df dr * r à 1 X 3X188 The solution fr *Y Nu to the problem (3.186), (3.187) and the de®nition (3.188) may be found, for instance, by a ®xed-point iterative procedure using a starting guess value for Nu D that is then successively improved. The ®nal result, Nu D 3X66Y 3X189 is in good agreement with the scale analysis result (3.169). Table 3.3 also gives the Nusselt numbers for several common types of ducts. 3.2.8 HEAT TRANSFER IN THE ENTRANCE REGION The previous results are valid for laminar internal forced ¯ow, when both velocity and temperature are fully developed, that is, for x b maxfX Y X T g. The length X T is that particular value of the x -coordinate where d T reaches the value of the hydraulic diameter. The scale analysis may be used to produce order of magnitude estimates, and Fig. 3.23a shows, qualitatively, the in¯uence of Pr on the scaling d T X T $D h . As seen previously, the ratio dad T increases monotonously with Pr; hence, X aX T has to vary conversely. Figure 3.23 The internal forced convection heat transfer in the Entrance Region for ¯uids with (a) Pr ( 1, (b) Pr ) 1. 534 Principles of Heat Transfer Heat Transfer For ¯uids with Pr ( 1, in virtue of (3.71), d T grows faster than d, d T x$x Pr À1a2 Re À1a2 x Y 3X190 and, if we consider that at the temperature entrance region limit x $ X T and d T $ D h , it follows that X T Pr À1a2 Re À1a2 X T $ D h Y or X T D h 1a2 Re D h Pr À1a2 $ 1X 3X191 Alternatively, this result may also be put in the form X T D h Re D h Pr À1 $ constY 3X192 where the constant value was identi®ed, empirically, as being ``approxi- mately 0.1.'' A similar scale analysis for the hydrodynamic problem leads to X T D h 1a2 Re À1a2 D h $ 1X 3X193 For ¯uids with Pr ) 1 (water, oils, etc.), d T $ a D h , although in the entrance region the temperature boundary layer grows more slowly than the velocity boundary layer. In this situation the velocity pro®le extends over D h (Fig. 3.23). Hence, in the temperature boundary layer the scale of u is U , and it may be shown that d T x$xRe À1a2 x Pr À1a2 , that is, a result that is identical to the one obtained for ¯uids with Pr ( 1. The scaling relations (3.193) and (3.191) lead to X T X $ PrY 3X194 a valuable, general conclusion that is valid for any Pr. The local Nusselt number in the thermally developing region x ( X T scales as Nu D hD h k $ q HH DT D h k $ D h d T $ xaD h Re D h Pr 23 À1a2 Y 3X195 and similarly to the d T (3.190) scale, it is valid for any Pr. This result was validated by other, more accurate solutions. 3.3 External Natural Convection Natural,orfree convection occurs when the ¯uid ¯ows ``by itself'' becauses of its density variation with the temperature, and not because of imposed, external means (e.g., a pump). For example, in a stagnant ¯uid reservoir that is in contact with a warmer, vertical wall (the heat source here), the ¯uid layer that contacts the wall is heated by the wall through thermal diffusion, and it becomes lighter as its density decreases. Consequently, this warmer ¯uid layer is entrained into a slow, upward motion that, provided the ¯uid 3. Convection Heat Transfer 535 Heat Transfer reservoir is large enough, does not perturb the ¯uid away from the wall. Because the hydrostatic pressure in the stagnant ¯uid reservoir decreases with altitude, a control volume of ¯uid conveyed in this ascending motion Ð in fact, a wall jet Ð expands while traveling upward. By virtue of the mass conservation principle (the reservoir contains a ®nite amount of ¯uid), the ¯uid control volume will eventually return to the bottom of the warm wall, entrained by a descending stream. In this closing sequence of its travel the ¯uid control volume is cooled and compressed (its density is increased) by the increasing hydrostatic pressure. Summarizing, the ¯uid control volume may be seen as a system that undergoes a cyclic sequence of heating, expansion, cooling, and compres- sion processes, which is in fact the classical thermodynamic work-producing cycle (Fig. 3.24) [2]. Here, unlike the classical thermodynamic cycles, the heating and expansion, on one hand, and the cooling and compression, on the other hand, are (respectively) simultaneous processes Ð neither at constant volume (the control volume expandsacompresses while heating upacooling down) nor isobaric (the hydrostatic pressure varies continuously with the altitude). The work potential produced by this cycle is ``consumed'' through internal friction between the ¯uid layers, which are in relative motion. In natural convection heat is transferred from the heat source (e.g., the warm, vertical wall) to the adjacent ¯uid layer by thermal diffusion, then by convection and diffusion within the ¯uid reservoir. When the ¯uid reservoir that freely convects the heat is external to the heat source, the convection heat transfer is called external. Figure 3.24 External natural convection: The ¯uid control volume acts as a system that undergoes a cyclic sequence of heating Ð the classical thermodynamic work-producing cycle. 536 Principles of Heat Transfer Heat Transfer 3.3.1 THE THERMAL BOUNDARY LAYER The ¯uid region where the temperature ®eld varies from the wall tempera- ture to the reservoir temperature and where, in fact, motion exists is called the thermal (temperature) boundary layer (Fig. 3.25). The central object of the thermal analysis is, again, to evaluate the heat transferred from the wall to the reservoir, and the bottom line to it is ®nding the heat transfer coef®cient h y def q HH wYy T w À T 0 À k dT dx x0 T w À T 0 X 3X196 Here T 0 is the reservoir temperature far away from the wall, T w is the wall temperature, k is the thermal conductivity of the ¯uid, and q HH wYy is the wall heat ¯ux rate in the y-direction (horizontal). The mass conservation equation is du dx dv dy 0Y 3X197 the momentum equation is r u du dx v du dy À dP dx m d 2 u dx 2 d 2 u dy 2 3X198 r u dv dx v dv dy À dP dy m d 2 v dx 2 d 2 v dy 2 À rgY 3X198 Figure 3.25 The structure of the natural convection boundary layer ¯ow: laminar, transition, and turbulent sections. 3. Convection Heat Transfer 537 Heat Transfer [...]... facing upwardY or cold surface facing downwardX @ 0X54Ra1a4 104 ` RaL ` 107 L Nu 0X15Ra1a4 107 ` RaL ` 109 X L Hot surface facing downwardY or cold surface facing upwardX NuL 0X27Ra1a4 105 ` RaL ` 101 0 X L Immersed bodies: Horizontal cylinder 27X W2 V b b b b b b b b b b 1a6 a ` 0X387RaD À5 12 NuD 0X6 4 9a16 58a27 b 10 ` RaD ` 10 Y any Pr b b b 0X559 b b b b b b 1 Y X Pr NuD HH... Nuy , de®ned based on the wall-averaged temperature difference Tw y À T0 , are recommended in ref 29: W Nuy 0X6Ray 1a5 a * * 3X248a laminarY 105 ` Ray ` 101 3 Nu 0X75RaÃ1a5 Y y y Nuy 0X568RaÃ0X22 y Nu y A 0X645RaÃ0X22 y * turbulentY 101 3 ` Ray ` 101 6 X 3X248b Nuy 0X55RaÃ1a5 laminar y Nuy 0X75RaÃ1a5 y turbulentX 3X248c 3X248d Churchill and Chu [27] proposed another correlation... below a certain critical value (Fig 3.26) Traditionally [4], the threshold limit is Ray $ 109 , regardless of the particular Pr number of the working ¯uid This criterion, which does not depend on Pr, was shown [26] to be better represented by the condition Gry $ 109 for all ¯uids within the range 10 3 ` Pr ` 103 $ $ 1 The velocity and temperature similarity pro®les shown in Fig 3.26 were computed... Y 3X260 kDT 0X2 Pr H for H aL P 2Y 10 , Pr ` 105 , RaH ` 101 3 In the opposite limit (the narrow cavity, LaH ` RaÀ1a4 ), if the enclosure is H tall enough, then the heat transfer between the side walls approaches the Heat Transfer 3.4.1 ENCLOSURES HEATED DIFFERENTIALLY FROM THE SIDES 552 Principles of Heat Transfer Table 3.4 Cross section NuaRaDh 1 106 X4 1 113X6 1 128 1 192 pure conduction regime... holds for 10 1 ` Ray ` 101 2 , and it covers the entire Rayleigh number $ $ range (laminar, transition, and turbulence) The physical properties that appear should be evaluated at the ®lm temperature, Tw T0 a2 For air Pr 0X72, (3.242) assumes the simpler form Nuy 0X825 0X325Ra1a6 2 X y 3X243 545 3 Convection Heat Transfer A more accurate correlation for the laminar regimes Gry ` 109 was... temperature difference, T w À TI For air Pr 0X72 this correlation predicts h i2 Nuy 0X825 0X328Ra1a6 Y 3X250 y which yields an asymptotic formula for high Rayleigh numbers Ray b 101 0 , namely, Nuy 0X107Ra1a6 X y 3X251 These relations may be rewritten in terms of the ¯ux Rayleigh number, *, *aNuy Ray by using the conversion Ray Ray 3.3.6 OTHER EXTERNAL NATURAL CONVECTION CONFIGURATIONS... (1975) Heat Transfer References 8 Fluid Dynamics NICOLAE CRACIUNOIU AND BOGDAN O CIOCIRLAN Department of Mechanical Engineering, Auburn University, Auburn, Alabama 36849 Inside 1 Fluids Fundamentals 2 Hydraulics 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1 .10 1.11 1.12 1.13 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2 .10 2.11 2.12 2.13 2.14 2.15 2.16 560 De®nitions 560 Systems of Units 560 Speci®c Weight 560 Viscosity... below put in terms of a critical Rayleigh number [38], which for shallow enclosures is RaH gb TH À Tc H 3 b $ 1708X an 3X263 NuH 0X069Ra1a3 Pr0X074 X H This expression is validated 3  105 ` RaH ` 7  109 [2] by experiments within the range 3.4.3 INCLINED ENCLOSURES HEATED DIFFERENTIALLY, FROM THE SIDES Figure 3.32 shows the impact of the orientation of the differentially heated cavity The recommended... t b b b b b 4 5 b b 1X6 b b b b  1 À 1708  sin 1X8t b b b RaL cos t b b b 4 b 1a3 5à b b b b RaL cos t b À1 Y b X 5830 t P 90 Y 180 t P t*Y 90 V ` t P 0 Y t* H X ` 10 L V t P 0 Y t* b ` b H b 10 X L Here, the quantities marked with * have to be set to zero when negative 555 References Figure 3.33 Natural convection ¯ow in the annular space between horizontal, coaxial cylinders... Heat Transfer Conf., Toronto, 1978, 6, 13±43 (1979) 37 S Kimura and A Bejan, The boundary layer natural convection regime in a rectangular cavity with uniform heat ¯ux from the side, J Heat Transfer 106 , 98 103 (1984) 38 A Pellew and R V Southwell, On maintained convection motion in a ¯uid heated from below, Proc Royal Soc A176, 312±343 (1940) 39 S Globe and D Dropkin, Natural convection heat transfer . 0X6Ra y * 1a5 Nu y 0X75Ra Ã1a5 y W a Y laminarY 10 5 ` Ra y * ` 10 13 3X248a Nu y 0X568Ra Ã0X22 y Nu y 0X645Ra Ã0X22 y A turbulentY 10 13 ` Ra y * ` 10 16 X 3X248b For air Pr 0X72, they. 0X328Ra 1a6 y hi 2 Y 3X250 which yields an asymptotic formula for high Rayleigh numbers Ra y b 10 10 , namely, Nu y 0X107Ra 1a6 y X 3X251 These relations may be rewritten in terms of the ¯ux Rayleigh. 0X492 Pr 9a16 45 8a27 V b b b b b ` b b b b b X W b b b b b a b b b b b Y 2 Y 3X242 that holds for 10 À1 $ ` Ra y $ ` 10 12 , and it covers the entire Rayleigh number range (laminar, transition, and turbulence).