cen29305_ch08.qxd 11/30/05 3:05 PM Page 451 INTERNAL FORCED CONVECTION CHAPTER 8 Liquid or gas flow through pipes or ducts is commonly used in heating

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9 cen29305_ch08.qxd 11/30/05 3:05 PM Page 451 INTERNAL FORCED CONVECTION CHAPTER 8 Liquid or gas flow through pipes or ducts is commonly used in heating and cooling applications. The fluid in such applications is forced to flow y a fan or pump through a flow section that is sufficiently long to accomplish the desired heat transfer. In this chapter we pay particular attention to the determination of the friction factor and convection coefficient since they are directly related to the pressure drop and heat transfer rate, respectively. These quantities are then used to determine the pumping power requirement and the required tue length. There is a fundamental difference etween external and internal flows. In external flow, considered in Chapter 7, the fluid has a free surface, and thus the oundary layer over the surface is free to grow indefinitely. In internal flow, however, the fluid is completely confined y the inner surfaces of the tue, and thus there is a limit on how much the oundary layer can grow. We start this chapter with a general physical description of internal flow, and the average velocity and average temperature. We continue with the discussion of the hydrodynamic and thermal entry lengths, developing flow, and fully developed flow. We then otain the velocity and temperature profiles for fully developed laminar flow, and develop relations for the friction factor and Nusselt numer. Finally we present empirical relations for developing and fully developed flows, and demonstrate their use. CONTENTS 8 1 Introduction Average Velocity and Temperature The Entrance Region General Thermal Analysis Laminar Flow in Tues Turulent Flow in Tues 473 Topic of Special Interest: Transitional Flow in Tues 482 Summary 490 References and Suggested Reading 491 Prolems 492 OBJECTIVES When you finish studying this chapter, you should e ale to: Otain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow, Have a visual understanding of different flow regions in internal flow, such as the entry and the fully developed flow regions, and calculate hydrodynamic and thermal entry lengths, Analyze heating and cooling of a fluid flowing in a tue under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference, Otain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt numer in fully developed laminar flow, and Determine the friction factor and Nusselt numer in fully developed turulent flow using empirical relations, and calculate the pressure drop and heat transfer rate. 451

10 cen29305_ch08.qxd 11/30/05 3:06 PM Page INTERNAL FORCED CONVECTION Then the logarithmic mean temperature difference and the rate of heat loss from the air ecome T i T e T ln 15.2 C ln T s T e ln T s T i Q ha s T ln (13.5 W/m 2 C)(6.4 m 2 )( 15.2 C) 1313 W Therefore, air will lose heat at a rate of 1313 W as it flows through the duct in the attic. Discussion The average fluid temperature is ( )/ C, which is sufficiently close to 80 C at which we evaluated the properties of air. Therefore, it is not necessary to re-evaluate the properties at this temperature and to repeat the calculations. TOPIC OF SPECIAL INTEREST Transitional Flow in Tues* Test section Inlet section 23.5 cm Flow 1.93 cm Re-entrant Flow Squared -edged Flow Bell-mouth Flow from calming section FIGURE 8 32 Schematic of the three differernt inlet configurations. An important design prolem in industrial heat exchangers arises when flow inside the tues falls into the transition region. In practical engineering design, the usual recommendation is to avoid design and operation in this region; however, this is not always feasile under design constraints. The usually cited transitional Reynolds numer range of aout 2300 (onset of turulence) to 10,000 (fully turulent condition) applies, strictly speaking, to a very steady and uniform entry flow with a rounded entrance. If the flow has a distured entrance typical of heat exchangers, in which there is a sudden contraction and possily even a re-entrant entrance, the transitional Reynolds numer range will e much different. Ghajar and coworkers in a series of papers (listed in the references) have experimentally investigated the inlet configuration effects on the fully developed transitional pressure drop under isothermal and heating conditions; and developing and fully developed transitional forced and mixed convection heat transfer in circular tues. Based on their experimental data, they have developed practical and easy to use correlations for the friction coefficient and the Nusselt numer in the transition region etween laminar and turulent flows. This section provides a rief summary of their work in the transition region. Pressure Drop in the Transition Region Pressure drops are measured in circular tues for fully developed flows in the transition regime for three types of inlet configurations shown in Fig. 8 32: re-entrant (tue extends eyond tuesheet face into head of *This section is contriuted y Professor Afshin J. Ghajar of Oklahoma State University.

11 cen29305_ch08.qxd 11/30/05 3:06 PM Page CHAPTER 8 distriutor), square-edged (tue end is flush with tuesheet face), and ellmouth (a tapered entrance of tue from tuesheet face) under isothermal and heating conditions, respectively. The widely used expressions for the friction factor f (also called the Darcy friction factor) or the friction coefficient C f (also called the Fanning friction factor) in laminar and turulent flows with heating are f lam 4C f, lam 4a 16 m Re am m s f tur 4C f, tur 4a m Re am 0.25 m s (8 79) (8 80) where the factors at the end account for the wall temperature effect on viscosity. The exponent m for laminar flows depends on a numer of factors while for turulent flows the most typically quoted value for heating is The transition friction factor is given as (Tam and Ghajar, 1997) f trans 4C f, trans 4c1 a Re B C A d a m m m s (8 81) where m m 1 m 2 Gr m 3 Pr m 4 (8 82) and the Grashof numer (Gr) which is a dimensionless numer representing the ratio of the uoyancy force to the viscous force is defined as Gr gd 3 (T s T )/n 2 (see Chapter 9 for more details). All properties appearing in the dimensionless numers C f,f, Re, Pr, and Gr are all evaluated at the ulk fluid temperature T. The values of the empirical constants in Eqs and 8 82 are listed in Tale 8 5. The range of application of Eq for the transition friction factor is as follows: Re-entrant: 2700 Re 5500, 16 Pr 35, 7410 Gr 158,300, 1.13 m /m s 2.13 Square-edged: 3500 Re 6900, 12 Pr 29, 6800 Gr 104,500, 1.11 m /m s 1.89 Bell-mouth: 5900 Re 9600, 8 Pr 15, 11,900 Gr 353,000, 1.05 m /m s 1.47 TABLE 8 5 Constants for transition friction coefficient correlation Inlet Geometry A B C m 1 m 2 m 3 m 4 Re-entrant Square-edged Bell-mouth

12 cen29305_ch08.qxd 11/30/05 3:06 PM Page INTERNAL FORCED CONVECTION C f,tur = Re C f,tur = Re C f,tur = Re 0.25 C f = f/ C f,lam = 16 / Re kw/m 2 Re-entrant Square-edged Bell-mouth C f = f/ C f,lam = 16 / Re kw/m 2 Re-entrant Square-edged Bell-mouth C f = f/ C f,lam = 16 / Re kw/m 2 Re-entrant Square-edged Bell-mouth FIGURE 8 33 Fully developed friction coeffficients for three different inlet configurations and heat fluxes (filled symols designate the start and end of the transition region for each inlet. (From Tam and Ghajar, 1997.) These correlations captured aout 82% of measured data within an error and of 10%, and 98% of measured data with 20%. For laminar flows with heating, Tam and Ghajar give the following constants for determining the exponent m in Eq. 8 79: m , m , m , and m , which is applicale over the following range of parameters: 1100 Re 7400, 6 Pr 36, 17,100 Gr 95,600, and 1.25 m /m s The fully developed friction coefficient results for the three different inlet configurations shown in Fig clearly estalish the influence of heating rate on the eginning and end of the transition regions, for each inlet configuration. In the laminar and transition regions, heating seems to have a significant influence on the value of the friction coefficient. However, in the turulent region, heating did not affect the magnitude of the friction coefficient. The significant influence of heating on the values of friction coefficient in the laminar and transition regions is directly due to the effect of secondary flow. The isothermal friction coefficients for the three inlet types showed that the range of the Reynolds numer values at which transition flow exists is strongly inlet-geometry dependent. Furthermore, heating caused an increase in the laminar and turulent friction coefficients and an increase in the lower and upper limits of the isothermal transition regime oundaries. The friction coefficient transition Reynolds numer ranges for the isothermal and nonisothermal (three different heating rates) and the three different inlets used in their study are summarized in Tale 8 6. TABLE 8 6 Transition Reynolds numers for friction coefficient Heat Flux Re-entrant Square-Edged Bell-Mouth 0 kw/m 2 (isothermal) 2870 Re Re Re kw/m Re Re Re kw/m Re Re Re kw/m Re Re Re 9560

13 cen29305_ch08.qxd 11/30/05 3:06 PM Page CHAPTER 8 Figure 8 34 shows the influence of inlet configuration on the eginning and end of the isothermal fully developed friction coefficients in the transition region. Note that the isothermal fully developed friction coefficients in the laminar, turulent, and transition regions can e otained easily from Eqs. 8 79, 8 80, and 8 81, respectively, y setting the exponent on the viscosity ratio correction to unity (i.e. with m 0). EXAMPLE 8 7 Nonisothermal Fully Developed Friction Coefficient in the Transition Region A tue with a ell-mouth inlet configuration is sujected to 8 kw/m 2 uniform wall heat flux. The tue has an inside diameter of m and a flow rate of m 3 /s. The liquid flowing inside the tue is ethylene glycoldistilled water mixture with a mass fraction of The properties of the ethylene glycol-distilled water mixture at the location of interest are Pr 11.6, n m 2 /s and m /m s Determine the fully developed friction coefficient at a location along the tue where the Grashof numer is Gr 60,800. What would the answer e if a square-edged inlet is used instead? C f = f/ C f,lam = 16 / Re Isothermal Re-entrant Square-edged Bell-mouth 2000 Re C f,tur = Re FIGURE 8 34 Influence of different inlet configurations on the isothermal fully developed friction coefficients (filled symols designate the start and end of the transition region for each inlet). (From Tam and Ghajar, 1997.) SOLUTION A liquid mixture flowing in a tue is sujected to uniform wall heat flux. The friction coefficients are to e determined for the ell-mouth and square-edged inlet cases. Assumptions Steady operating conditions exist. Properties The properties of the ethylene giycol-distilled water mixture are given to e Pr 11.6, n m 2 /s and m /m s Analysis For the calculation of the nonisothermal fully developed friction coefficient, it is necessary to determine the flow regime efore making any decision regarding which friction coefficient relation to use. The Reynolds numer at the specified location is Re (V# /A c )D n since [( m 3 /s)/( m 2 )]( m) m 2 /s A c pd 2 /4 p( m) 2 / m 2 From Tale 8 6, we see that for a ell-mouth inlet and a heat flux of 8 kw/m 2 the flow is in the transition region. Therefore, Eq applies. Reading the constants A, B, and C and m 1, m 2, m 3, and m 4 from Tale 8 5, the friction coefficient is determined to e C f,trans c1 a Re B C A d a m m m s c1 a d (1.14) ,

14 cen29305_ch08.qxd 11/30/05 3:06 PM Page INTERNAL FORCED CONVECTION Square-Edged Inlet Case For this inlet shape, the Reynolds numer of the flow is the same as that of the ell-mouth inlet (Re 7651). However, it is necessary to check the type of flow regime for this particular inlet with 8 kw/m 2 of heating. From Tale 8 6, the transition Reynolds numer range for this case is 3860 Re 5200, which means that the flow in this case is turulent and Eq is the appropriate equation to use. It gives C f,tur a m Re am a m s (1.14) 0.25 Discussion Note that the friction factors f can e determined y multiplying the friction coefficient values y 4. Heat Transfer in the Transition Region Ghajar and coworkers also experimentally investigated the inlet configuration effects on heat transfer in the transition region etween laminar and turulent flows in tues for the same three inlet configurations shown in Fig They proposed some prediction methods for this regime to ridge etween laminar methods and turulent methods, applicale to forced and mixed convection in the entrance and fully developed regions for the three types of inlet configurations, which are presented next. The local heat transfer coefficient in transition flow is otained from the transition Nusselt numer, Nu trans, which is calculated as follows at a distance x from the entrance: Nu trans Nu lam {exp[(a Re)/] Nu c tur} c (8 83) where Nu lam is the laminar flow Nusselt numer for entrance region laminar flows with natural convection effects, 1/3 Re Pr D Nu lam 1.24ca 0.025(Gr Pr) 0.75 d a m 0.14 x m s (8 84) and Nu tur is the turulent flow Nusselt numer with developing flow effects, TABLE 8 7 Constants for transition heat transfer correlation Inlet Geometry a c Re-entrant Square-edged Bell-mouth Nu tur 0.023Re 0.8 Pr a x D a m 0.14 m s (8 85) The physical properties appearing in the dimensionless numers Nu, Re, Pr, and Gr all are evaluated at the ulk fluid temperature T. The values of the empirical constants a,, and c in Eq depend on the inlet configuration and are given in Tale 8 7. The viscosity ratio accounts for the temperature effect on the process. The range of application of the heat transfer method ased on their dataase of 1290 points (441 points for re-entrant

15 cen29305_ch08.qxd 11/30/05 3:06 PM Page CHAPTER 8 inlet, 416 for square-edged inlet and 433 points for ell-mouth inlet) is as follows: Re-entrant: 3 x/d 192, 1700 Re 9100, 5 Pr 51, 4000 Gr 210,000, 1.2 m /m s 2.2 Square-edged: 3 x/d 192, 1600 Re 10,700, 5 Pr 55, 4000 Gr 250,000, 1.2 m /m s 2.6 Bell-mouth: 3 x/d 192, 3300 Re 11,100, 13 Pr 77, 6000 Gr 110,000, 1.2 m /m s 3.1 These correlations capture aout 70% of measured data within an error and of 10%, and 97% of measured data with 20%, which is remarkale for transition flows. The individual expressions aove for Nu lam and Nu tur can e used alone for developing flows in those respective regimes. The lower and upper limits of the heat transfer transition Reynolds numer ranges for the three different inlets are summarized in Tale 8 8. The results shown in this tale indicate that the re-entrant inlet configuration causes the earliest transition from laminar flow into the transition regime (at aout 2000) while the ell-mouth entrance retards this regime change (at aout 3500). The square-edged entrance falls in etween (at aout 2400), which is close to the often quoted value of 2300 in most textooks. Figure 8 35 clearly shows the influence of inlet configuration on the eginning and end of the heat transfer transition region. This figure plots the local average peripheral heat transfer coefficients in terms of the Colurn j factor ( j H St Pr 0.67 ) versus local Reynolds numer for all flow regimes at the length-to-diameter ratio of 192, and St is the Stanton numer, which is also a dimensionless heat transfer coefficient (see Chapter 6 for more details), defined as St Nu/(Re Pr). The filled symols in Fig represent the start and end of the heat transfer transition region for each inlet configuration. Note the large influence of natural convection superimposed on the forced convective laminar-flow heat transfer process (Nu for a fully developed laminar flow with a uniform heat-flux oundary condition without uoyancy effects), yielding a mixed convection value of aout Nu Equation 8 84 includes this uoyancy effect through the Grashof numer. In a susequent study, Tam and Ghajar (1998) experimentally investigated the ehaviour of local heat transfer coefficients in the transition region for a tue with a ell-mouth inlet. This type of inlet is used in some heat exchangers mainly to avoid the presence of eddies which are elieved to e one of the causes for erosion in the tue inlet region. For the St Pr Re-entrant Square-edged Bell-mouth St tur Pr 0.67 = 0.023Re 0.2 (m /m s ) Re St lam Pr 0.67 = 4.364Re 1 Pr FIGURE 8 35 Influence of different inlets on the heat transfer transition region at x/d 192 (filled symols designate the start and end of the transition region for each inlet) etween limits of Dittus Boelter correlation (Nu Re 0.8 Pr n ) for fully developed turulent flow (using n 1/3 for heating) and Nu for fully developed laminar flow with a uniform heat flux oundary condition. Note uoyancy effect on the laminar flow data giving the much larger mixed convection heat transfer coefficient. (From Ghajar and Tam, 1994.) TABLE 8 8 The lower and upper limits of the heat transfer transition Reynolds numers Inlet Geometry Lower Limit Upper Limit Re-entrant Re lower [192 (x/d)] Re upper [192 (x/d)] Square-edged Re lower [192 (x/d)] Re upper [192 (x/d)] Bell-mouth Re lower [192 (x/d)] Re upper [192 (x/d)]

16 cen29305_ch08.qxd 11/30/05 3:06 PM Page INTERNAL FORCED CONVECTION Nu Nu Nu Re-entrant transition region Re = 4720~6020 Squar-edged transition region Re = 4170~ Bell-mouth transition region 0 Re = 4990~ x/d FIGURE 8 36 Variation of local Nusselt numer with length for the re-entrant, squareedged, and ell-mouth inlets in the transition region. (From Tam and Ghajar, 1998.) ell-mouth inlet, the variation of the local heat transfer coefficient with length in the transition and turulent flow regions is very unusual. For this inlet geometry, the oundary layer along the tue wall is at first laminar and then changes through a transition to the turulent condition causing a dip in the Nu versus x/d curve. In their experiments with a fixed inside diameter of mm, the length of the dip in the transition region was much longer (100 x/d 175) than in the turulent region (x/d 25). The presence of the dip in the transition region causes a significant influence in oth the local and the average heat transfer coefficients. This is particularly important for heat transfer calculations in short tue heat exchangers with a ellmouth inlet. Figure 8 36 shows the variation of local Nusselt numer along the tue length in the transition region for the three inlet configurations at comparale Reynolds numers. EXAMPLE 8 8 Heat Transfer in the Transition Region Ethylene glycol-distilled water mixture with a mass fraction of 0.6 and a flow rate of m 3 /s flows inside a tue with an inside diameter of m sujected to uniform wall heat flux. For this flow, determine the Nusselt numer at the location x/d 90 if the inlet configuration of the tue is: (a) re-entrant, () square-edged, and (c) ell-mouth. At this location, the local Grashof numer is Gr 51,770. The properties of ethylene glycol-distilled water mixture at the location of interest are Pr 29.2, m 2 /s and m /m s SOLUTION A liquid mixture flowing in a tue is sujected to uniform wall heat flux. The Nusselt numer at a specified location is to e determined for three different tue inlet configurations. Assumptions Steady operating conditions exist. Properties The properties of the ethylene glycol-distilled water mixture are given to e Pr 29.2, n m 2 /s and m /m s Analysis For a tue with a known diameter and volume flow rate, the type of flow regime is determined efore making any decision regarding which Nusselt numer correlation to use. The Reynolds numer at the specified location is since Re (V# /A c )D v [( m 3 /s)( m 2 )]( m) m 2 /s A c pd 2 /4 p( m) 2 / m 2 Therefore, the flow regime is in the transition region for all three inlet configurations (thus use the information given in Tale 8 8 with x/d 90) and therefore Eq should e used with the constants a,, c found in Tale 8 7. However, Nu lam and Nu tur are the inputs to Eq and they need to e evaluated first from Eqs and 8 85, respectively. It should e mentioned that the correlations for Nu lam and Nu tur have no inlet dependency.

17 cen29305_ch08.qxd 11/30/05 3:06 PM Page CHAPTER 8 From Eq. 8 84: 1/3 Re Pr D Nu lam 1.24ca 0.025(Gr Pr) 0.75 d a m 0.14 x m s 1.24ca (6714)(29.2) 1/ [(51,770)(29.2)] 0.75 d (1.77) From Eq. 8 85: Nu tur 0.023Re 0.8 Pr a x D a m 0.14 m s Then the transition Nusselt numer can e determined from Eq. 8 83, Case 1: For re-entrant inlet: Nu trans 19.9 {exp[( )/276] } Case 2: For square-edged inlet: Nu trans 19.9 {exp[( )/207] } Case 3: For ell-mouth inlet: 0.023(6714) 0.8 (29.2) (90) (1.77) c Nu trans Nu lam {exp[(a Re)/] Nu tur } c Nu trans 19.9 {exp[( )/237] } Discussion It is worth mentioning that, for the re-entrant and square-edged inlets, the flow ehaves normally. For the ell-mouth inlet, the Nusselt numer is low in comparison to the other two inlets. This is ecause of the unusual ehaviour of the ell-mouth inlet noted earlier (see Fig. 8 36); i.e., the oundary layer along the tue wall is at first laminar and then changes through a transition region to the turulent condition. REFERENCES 1. A. J. Ghajar and K. F. Madon. Pressure Drop Measurements in the Transition Region for a Circular Tue with Three Different Inlet Configurations. Experimental Thermal and Fluid Science, Vol. 5 (1992), pp A. J. Ghajar and L. M. Tam. Heat Transfer Measurements and Correlations in the Transition Region for a Circular Tue with Three Different Inlet Configurations. Experimental Thermal and Fluid Science, Vol. 8 (1994), pp A. J. Ghajar and L. M. Tam. Flow Regime Map for a Horizontal Pipe with Uniform Wall Heat Flux and Three Inlet Configurations. Experimental Thermal and Fluid Science, Vol. 10 (1995), pp

18 cen29305_ch08.qxd 11/30/05 3:06 PM Page INTERNAL FORCED CONVECTION 4. A. J. Ghajar, L. M. Tam, and S. C. Tam. Improved Heat Transfer Correlation in the Transition Region for a Circular Tue with Three Inlet Configurations Using Artificial Neural Networks. Heat Transfer Engineering, Vol. 25, No. 2 (2004), pp L. M. Tam and A. J. Ghajar. Effect of Inlet Geometry and Heating on the Fully Developed Friction Factor in the Transition Region of a Horizontal Tue. Experimental Thermal and Fluid Science, Vol. 15 (1997), pp L. M. Tam and A. J. Ghajar. The Unusual Behavior of Local Heat Transfer Coefficient in a Circular Tue with a Bell-Mouth Inlet. Experimental Thermal and Fluid Science, Vol. 16 (1998), pp SUMMARY Internal flow is characterized y the fluid eing completely confined y the inner surfaces of the tue. The mean or average velocity and temperature for a circular tue of radius R are expressed as 2 2 V avg u(r)rdr and T m u(r)t(r)rdr V avg R R R R The Reynolds numer for internal flow and the hydraulic diameter are defined as rvavg D V avg D Re and D h m The flow in a tue is laminar for Re 2300, turulent for aout Re 10,000, and transitional in etween. The length of the region from the tue inlet to the point at which the oundary layer merges at the centerline is the hydrodynamic entry length L h. The region eyond the entrance region in which the velocity profile is fully developed is the hydrodynamically fully developed region. The length of the region of flow over which the thermal oundary layer develops and reaches the tue center is the thermal entry length L t. The region in which the flow is oth hydrodynamically and thermally developed is the fully developed flow region. The entry lengths are given y L h, laminar 0.05 Re D L t, laminar 0.05 Re Pr D Pr L h, laminar L h, turulent L t, turulent 10D For q s constant, the rate of heat transfer is expressed as For T s constant, we have Q q s A s m c p (T e T i ) 4A c p Q ha s T ln m c p (T e T i ) T e T s (T s T i )exp( ha s /m c p ) T i T e T e T i T ln ln[(t s T e )/(T s T i )] ln( T e / T i ) The irreversile pressure loss due to frictional effects and the required pumping power to overcome this loss for a volume flow rate of V are rv 2 avg L P L f and W pump V P D 2 L For fully developed laminar flow in a circular pipe, we have: 2 r u(r) 2V avg 1 R 2 64m f 64 rdv avg Re u max 1 r 2 R 2 PR V V avg A c 2 pr 2 pr 4 P 8mL 8mL pr 4 P 128mL Circular tue, laminar (q s constant): hd Nu 4.36 k Circular tue, laminar (T s constant): hd Nu 3.66 k For developing laminar flow in the entrance region with constant surface temperature, we have 0.065(D/L) Re Pr Circular tue: Nu [(D/L) Re Pr] 2/3

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