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Combined free and forced convection in a horizontal tube under uniform heat flux Kupper, Arthur K. 1968

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COMBINED FREE AND FORCED CONVECTION IN A HORIZONTAL TUBE UNDER UNIFORM HEAT FLUX by ARTHUR K. KUPPER di p l . Ing. ETH 1964 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY oft BRITISH COLUMBIA October 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n D e p a r t m e n t o f A/^//. ^AJ<6, The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, C a n a d a D a t e /Z-Z/—&<8 i i ABSTRACT This thesis presents experimental results of combined free aid forced convection laminar heat transfer for water flowing through a c i r -cular horizontal tube with uniform w a l l heat f l u x . The Reynolds number ranged from 100 to 2000, and changes i n heat transfer rate allowed a v a r i -ation, of Grashof number from 300 to 30,000. The Prandtl number ranged from 4 to 9. The effect of secondary flow created by free convection occurring at higher Grashof number indicates an increase i n Nusselt number up to 200 per cent. For the fully-developed region two tentative corre-lations are given. The expression Nu = 48/11 + 0.047 P r 1 / 3 (Re Ra) 1/ 5 correlates 53 per cent of the data to within ± 10 per cent. Another s l i g h t l y more accurate expression which correlates 68 per cent of the data to within ± 10 per cent, but does not s a t i s f y the pure forced con-vection, i s Nu = 2.41 + 0.082 P r 1 / 3 (Re Ra ) 1 / 5 ACKNOWLEDGEMENTS I wish to express my deep gratitude to Dr. M. Iqbal and to Dr. E.G. Hauptmann for t h e i r guidance and advice throughout my graduate studies. Also, I wish to thank the members of the Mechanical Engineering Depart-ment s t a f f for t h e i r assistance and the Department for the use of i t s f a c i l i t i e s . The f i n a n c i a l assistance given by the University of B r i t i s h Columiba and by the National Research Council of Canada i s g r a t e f u l l y acknowledged. i v TABLE OF CONTENTS ABSTRACT i i ACKNOWLEDGEMENT i i i • LIST OF FIGURES v i NOMENCLATURE v i i i INTRODUCTION 1 LITERATURE SURVEY 4 Uniform Wall Temperature 5 Uniform Wall Heat Flux 9 PURPOSE OF EXPERIMENT 19 DESCRIPTION OF EXPERIMENTAL EQUIPMENT 19 Flow Loop 19 Test Section 21 Test Section Insulation 25 Power Supply 27 Temperature Measurement 27 EXPERIMENTAL PROCEDURE .29 Test Procedure 29 Thermocouple Calibration Method 29 Calibration Result 30 RESULTS AND DISCUSSION 31 Entrance Length and Circumferential Temperature Variation 32 A x i a l Temperature P r o f i l e s 35 V Contents continued Local Nusselt Number . 38 Heat Transfer Correlation 47 CONCLUSION 51 REFERENCES 52 APPENDIX A DERIVATION OF GOVERNING EQUATIONS 55 APPENDIX B LOCATION OF TEST SECTION THERMOCOUPLES 56 APPENDIX C SAMPLE CALCULATION 58 APPENDIX D ERROR ANALYSIS 62 APPENDIX E TABULATED LOCAL HEAT TRANSFER DATA 65 LIST OF FIGURES l a Velocity P r o f i l e s for Heatea upward Flow lb Motion of Flui d P a r t i c l e s under Buoyancy Effects 2 Coordinate System 3 Temperature Variation i n the A x i a l Direction 4 Reproduction of Figure 1 from Reference 14 5 Reproduction of Figure 3 from Reference 14 6 Reproduction of Figure 4 from Reference 14 7 Reproduction of Figure 9 from Reference 16 8 Reproduction of Figure 6 from Reference 18 9 Schematic of Heat Transfer Loop 10 P a r t i a l View of Experimental Arrangement 11 View of Test Section 12 Thermocouple Placing on Test Section Circumference 13 Schematic of Power Supply 14 Thermocouple C i r c u i t 15 Representative Temperature P r o f i l e , Run Number 3 16 Representative Temperature P r o f i l e , Run Number 42 17 Representative Temperature P r o f i l e , Run Number 1 18 Representative Temperature P r o f i l e , Run Number 43 19 • Representative Temperature P r o f i l e , Run Number 37 v i i L i s t of Figures continued 20 Representative Temperature P r o f i l e , Run Number 27 40 21 Representative Nusselt Number P r o f i l e , Run Number 3 41 22 Representative Nusselt Number P r o f i l e , Run Number 37 43 23 Representative Nusselt Number P r o f i l e , Run Number 1 44 24 Representative Nusselt Number P r o f i l e , Run Number 43 45 25 Representative Nusselt Number P r o f i l e s , Run Numbers 1 and 43 46 26 Correlation of Nusselt Number, Equation 17 48 27 Correlation of Nusselt Number, Equation 18 49 v i i i NOMENCLATURE A A x i a l temperature gradient a Tube radius C A x i a l pressure gradient i n f l u i d c Specific heat at constant pressure P D Tube diameter E Voltage 8 E AT D3 Gr — 6 — 5 . Grashof number Gz -r- Re Pr -7^ - , Graetz number 4 L2 g Acceleration of gravity h T~^ T ' n e a t t r a n s r e r c o e f f i c i e n t w k Thermal conductivity L Characteristic length m Mass flow rate Nu • Nusselt number k P Power input Pr — f — , Prandtl number k q Wall heat f l u x density R Dimensionless r a d i a l coordinate i x Nomenclature continued r Radial coordinate Ra. Gr Pr , Rayleigh number Re " , Reynolds number T Fl u i d bulk temperature T w Tube temperature T* Dimensionless temperature u Average v e l o c i t y Dimensionless a x i a l v e l o c i t y v^ Velocity i n r a d i a l d i r e c t i o n V Q Velocity i n angular d i r e c t i o n X Coordinate i n a x i a l d i r e c t i o n a Tube i n c l i n a t i o n with respect to the horizontal position 3 Expansion c o e f f i c i e n t AT Temperature d i f f e r e n c e ^ f l u i d bulk to wal l •V Laplacian 8 Angular coordinate V Dynamic v i s c o s i t y of f l u i d V Kinematic v i s c o s i t y of f l u i d p Density of f l u i d i|) Stokes stream function 1 INTRODUCTION The rate of heat transfer between a s o l i d surface and a f l u i d may depend on both free and forced convection mechanisms. Forced convection occurs when f l u i d motion i s produced by a pump or s i m i l a r means. In free or natural convection, f l u i d motion occurs due to buoyancy forces pro-duced from temperature (or equivalently density) differences a r i s i n g from the heat transfer s i t u a t i o n i t s e l f . For laminar heat transfer asso-ciated with a c i r c u l a r tube, the buoyancy forces are characterized by the dimensionless Grashof number, and the forced flow i s characterized by the Reynolds number. In most physical s i t u a t i o n s , both modes of heat transfer are present, and t h e i r r e l a t i v e magnitude determines whether a flow i s considered as forced, free, or combined free and forced convection. The present investigation originates from study of flow and heat transfer through tubes of f l a t plate solar c o l l e c t o r s , where both free and forced convection mechanisms are important. In practice, pure forced convection i s never exactly r e a l i z e d , and use of the term merely implies that free convection effects are n e g l i g i b l e . Where natural convection effects are not n e g l i g i b l e , the s o l i d surface orientation with respect to the dire c t i o n of gravitational force also be-comes important. For heating of upward flow i n v e r t i c a l tubes, free convection aids the forced upward motion of f l u i d near the w a l l . This creates steeper velocity and temperature gradients near the wall and hence results i n a higher heat transfer c o e f f i c i e n t . To s a t i s f y continuity, the increased ve l o c i t y near the w a l l results i n flow retardation at the tube centre, 2 and in extreme cases a reversal of flow at the tube centre may even occur (Fig. l a ) . Free convection opposes the forced flow during heating of fluids flowing downward, and a decrease i n heat transfer rate results. In the case of a horizontal tube, the free and forced convectlve mo-tions are perpendicular to one another. For pure forced flow the veloci-ty profile i s symmetrical with respect to the tube axis. However, when the tube wall i s heated, the f l u i d particles receive heat from the wall, forming a temperature gradient through the f l u i d . The f l u i d particles near the wall have a lower density (in most fluids) than those in the centre of the tube, and motion i s created because of gravity forces. The fl u i d particles near the wall rise upwards, while those in the centre move downwards. Superimposing this motion on forced flow, the f l u i d par-t i c l e s develop spi r a l trajectories, ascending near the wall toward the top and descending i n the centre as they move through the tube (Fig. l b ) . There are many possible variations of the problem stated. The f l u i d may be cooled or heated from the tube wall i n different ways. In addi-tion, internal heat sources may also be present. The two most interest-ing and practical cases of combined free and forced convection i n hori-zontal tubes occur when either the wall temperature or wall heat flux are approximately uniform. The present investigation deals with the latter case. LITERATURE SURVEY Forced convection is independent of tube orientation and has been extensively investigated. The influence of free convection on forced flow in horizontal tubes has been given soraev?hat less attention. Five equations govern the temperature and velocity fields in this problem. The continuity equation, the momentum equations (one for each of the three velocity components) and the energy equation. Using cylindrical coordinates, as outlined in Fig. 2,. and nondimensional forms Iqbal and Stachiewicz [12]* obtained for angles a other than 90°, ' X + i ' l f e - f f!> V x + We - RaT* s i n „ - 0 R 3R 36 36 3RJ x The derivation of the above equations is explained i n Appendix A. FIGURE 2 Coordinate System l« * Numbers in square brackets refer to references. 5 In the above equation physical properties of the f l u i d are considered constant except for density variation affecting the buoyancy terms. In the energy equation, axial conduction, viscous dissipation and pressure work terms have been ignored. Furthermore, the flow is considered hydro-dynamically fully-developed. Two boundary conditions for the temperature at the wall are usually encountered in most practical instances of laminar heat transfer in c i r -cular horizontal tubes: uniform wall temperature (such as arising from heating with a condensing vapor), and uniform wall heat flux (such as arising from heating with electric currents). These two boundary condi-tions produce axial temperature variations as shown in Fig. 3. a) Uniform Wall Temperature Apparently no theoretical study has yet been made for combined free and forced convection in horizontal tubes with constant wall temperature. There are, however, several reports regarding experimental investigations. Colburn [1] has shown that natural convection may increase the rate of heat transfer by a factor of three or four. He presented the rela-tionship Nu = 1.75 Gz 1/ 3 (1 + 0.015 Gr */3) ( u . / u j 1 / 3 (4) am i b r where Nu i s the Nusselt number with f l u i d properties based on the arith-am r r metic mean temperature difference between f l u i d and tube (the indices " f " and "b" refer to film and bulk temperature respectively). The correction factor (1 + 0.015 Gr, 1/ 3) (u, /u^) 1^ 3 allows for both natural convection DISTANCE ALONG TUBE (3A) WALL TEMPERATURE DISTANCE ALONG TUBE (3B) FIGURE 3 TEMPERATURE VARIATION IN THE AXIAL DIRECTION 5A CONSTANT DALL MEAT FLUX 3B CONSTANT WALL TEMPERATURE 7 and r a d i a l v a r i a t i o n of f l u i d properties. Sleder and Tate [2] suggested that the v i s c o s i t y v a r i a t i o n i n equa-t i o n (4) might be replaced by ) 0 , 1 1 * , where u denotes the f l u i d v i s -b w w cosity evaluated at the w a l l temperature. They also suggested the Gras-hof number should be evaluated at the mean f l u i d temperature rather than f i l m temperature. However, these changes throw no further l i g h t on the natural convection process i t s e l f . From t h e i r extensive experimental data, Kern and Othmer [3] noticed that Sieder and Tate's [2] equation predicted high heat transfer c o e f f i -cients at high Reynolds numbers, and the opposite at low Reynolds numbers. For t h i s reason they introduced an arbitrary function ( l n 1 0 R e ) - 1 i n the natural convection term, obtaining the equation Kern and Othmer obtained markedly better corr e l a t i o n of t h e i r results with t h i s equation, than by using the preceeding equations. M a r t i n e l l i et a l . [4] presented a t h e o r e t i c a l approach to the prob-lem of combined forced and free convection. The analysis applied to the cases of heating of f l u i d s flowing v e r t i c a l l y upwards, and cooling of f l u i d s flowing downwards i n tubes with uniform w a l l temperature. They obtained the equation Nu = 1.86 { (f) (-Sf) A } am u K. L, Nu = 1.75 Ei { Gz + 0.0722 F 2 (• a m 1 m N (6) The correction factor F i allows for use of arithmetic rather than loga-8 rithmic mean temperature d i f f e r e n c e s , while the correction factor F 2 allows f o r a reduction i n free convection forces as the f l u i d bulk temp-erature r i s e s . From data f o r heating of o i l s i n ho r i z o n t a l tubes, Eubank and Proc-tor [5] modified equation (6) to the form Gr Pr D 0 • 1 + 0 1/3 „ 0 . 1 4 Nu = 1.75 . Gz + 12.6 , m m \ . , V , am { m ( ) } (—) (7) L Ww The Grashof number i s now based on mean temperature difference between the w a l l and f l u i d , and the properties of the f l u i d are based on the mean bulk temperature. McAdams [6] recommended an equation s i m i l a r to that presented by Eubank and Proctor [5], except with the c o e f f i c i e n t s 12.6 and 0.4 re-placed by 0.04 and 0.75 re s p e c t i v e l y . Jackson, Spurlock, and Purdy [7] have studied heat t r a n s f e r rates using a i r i n a constant temperature h o r i z o n t a l tube. Their r e s u l t s are we l l represented by the semi-theoretical equation Nu, = 2.67 { Gz 2 + 0.00872 (Gr Pr ) T - 5 } 1/ 6 (8) 1 m m w w The subscript "lm" denotes that the heat trans f e r c o e f f i c i e n t i s based on logarithmic mean temperature d i f f e r e n c e . O l i v e r [8] reported an i n v e s t i g a t i o n of the same problem using re-l a t i v e l y non-viscous l i q u i d s . He presented the following r e l a t i o n s h i p T 0 . 7 1/3 „ ,, Nu = 1.75 { Gz + 5.6 • IOT1* (Gr Pr } (u / \ i j ^ ' l k (9) am m m D w b I t should be noted that the r a t i o L/D rather than D/L i s used. The rea-son i s that when the Grashof number i s multiplied by the D/L r a t i o , a term D1* occurs, with the result that a small v a r i a t i o n i n D produces large variations i n (Gr Pr 5). This v a r i a t i o n i s not reflected by corre-sponding changes i n heat transfer c o e f f i c i e n t s . Brown and Thomas [9] conducted a study of combined free and forced convection heat transfer i n horizontal tubes, and proposed the r e l a t i o n Nu ( u / u , )°'lL* = 1.75 { Gz + 0.012 (Gz G r 1 / 3 ) 4 / 3 } l / 3 (10) W D However, t h e i r results do not agree with e x i s t i n g correlations. Martin and Carmichael [10] have also reported results for l o c a l heat transfer c o e f f i c i e n t s for th i s problem, using water as the heat transfer medium. b) Uniform Wall Heat Flux Very few theo r e t i c a l analyses are available for combined free and forced convection i n a horizontal tube under uniform heat f l u x . Morton [11] analysed the problem by solving equations (1) to (3) by a perturbation method, using the Rayleigh number as a perturbation par-ameter. The Nusselt number based on the mean temperature was evaluated from the r e s u l t i n g d i s t r i b u t i o n s and was expressed by the following asymp-t o t i c expression v a l i d for small values of Ra Re and Pr Nu = 6 { 1 + (0.0586 - 0.0852 Pr +0.2686 P r 2 ) ( f fof^) 2 + .....} (11) The mean temperature was defined as T = / T dA / A. No numerical'limits 10 of Rayleigh number were given for the region of v a l i d i t y of the r e s u l t s . Iqbal and Stachiewicz [12] analyzed the problem including tht effect of tube o r i e n t a t i o n . They also solved equation (1) - (3) by a perturba-t i o n method using the Rayleigh number as a perturbation parameter. How-ever, the Nusselt number was based on the bulk temperature, defined as T b - / T v d A / / v d A (12) They have shown that for horizontal tubes, the Nusselt number based on equation (12) i s a function of the product RaRe as well as Pr, while f o r flow through i n c l i n e d tubes Nu = f (Re, Ra, P r ) . I t i s known [20] that for flow through v e r t i c a l tubes, Nu * f (Ra) only. Another t h e o r e t i c a l investigation of combined free and forced con-vectioh i n horizontal tubes with uniform heat f l u x was reported by Del Casal and G i l l [13]. In t h e i r analysis they assumed the f l u i d properties constant, except that density was allowed to vary throughout the govern-ing equations. The governing equations were also solved by perturbation analysis, the r a t i o Gr/Re2 being used as perturbation parameter. Del Casal and G i l l state that the Froude number, characterizing density differences, also enters the problem. However, no solution for the energy equation or Nusselt number was given. There are few experimental investigations available i n the l i t e r a -ture for the case of combined free and forced convection i n horizontal tubes under uniform heat f l u x . Ede [14] carried out an experiment using seven di f f e r e n t tube dian-11 eters, with both a i r and water used as the heat transfer medium. Rey-nolds numbers were varied from 300 to 100,000, while Grashof number var-ied up vo 10 7. The Prandtl number was approximately 0.7 for a i r , and varied from 4 - 1 2 for water. Ede plotted his results i n three d i f f e r e n t ways, and his figures are reproduced i n Figs. 4 to 6. Unfortunately, no mention of the values of Grashof number employed at various Reynolds num-bers was made. Ede also presented an empirical c o r r e l a t i o n for the Nus-s e l t number as „ Nu = 4.36 (1 + 0.06 Gr 0' 3) (13) Considering the t h e o r e t i c a l studies of combined free and forced convec-t i o n i n horizontal tubes, one notices the lack of parameters such as Prandtl number and Reynolds number i n Ede's cor r e l a t i o n . McComas and Eckert [15] reported an experimental study on combined free and forced convection heat transfer under uniform w a l l heat f l u x i n a horizontal tube. A i r was used as heat transfer medium. The Grashof num-ber was varied by using a i r at different pressures. I t ranged from 1 to 1000. McComas and Eckert did not present any working c o r r e l a t i o n , on the grounds that i n s u f f i c i e n t experimental data was available for t h i s s i t u a -t i o n . Recently, a report was published by Mori et a l . [16] on free and forced convection i n horizontal tubes with constant w a l l heat f l u x , using a i r as a heat transfer medium. Measurements of temperature and v e l o c i t y d i s t r i b u t i o n at certain cross sections of the test tube were made, and the l o c a l Nusselt number was calculated from l o c a l w a l l temperature. The 12 lOOOr IOO 10 AIR o WATER . if ,ki<\ IOO iOOO lOOOO lOOOOO FIGURE 4 REPRODUCTION OF FIGURE 1 FROM REFERENCE 14 1 3 I 40-Nu 30 20 IO 8 7 6 i j — I : i Gr «= IOS j v.' . • * i •: • • j | • • • * • • s • . « • i i • i i 1 " ' i ! ' i I ; i • ! ! i l l i ' i ' i i l l 5 6 7 S 9 iO3 3 4 /?<? — - . 5 6 FIGURE 5 REPRODUCTION OF FIGURE 3 FROM REFERENCE 14 4 C 3C 2d lOf-- THEORY 67- • O 10s IO7 FIGURE 6 REPRODUCTION OF FIGURE 4 FROM REFERENCE 14 14 Reynolds number was varied from 100 to 130,000. The Nusselt number cor-r e l a t i o n was expressed by the r e l a t i o n which represents curve 2 of Fig. 7. I t remains to be explained why th i s form of correlation was chosen, since one notices that equation (14) i s equivalent to Although Morton's analysis shows that Nusselt number i s a function of the RaRe product, as wel l as Pr, Mori's equation does not contain the Prandtl number e x p l i c i t l y . The reason for t h i s appears to be due to the fact that Mori used only a i r (Pr=l) i n his experimental investigation and therefore no correlation with Pr could be attempted. In a second report, Mori and Futagami [17] present a th e o r e t i c a l i n -vestigation of the same problem. In th e i r analysis they divide the flow i n the tube into a thin layer along the tube w a l l and a core region. In the thin layer, the ve l o c i t y and temperature f i e l d s are affected by v i s -cosity and thermal conductivity. On the other hand, i n the core region, the v e l o c i t y and temperature f i e l d s are affected mainly by the secondary flow and the effects of v i s c o s i t y and thermal conductivity are disregarded. They give an approximate solution for very large products of ReRa, and ex-pressed the Nusselt number as Nu = 0.61 (ReRa) 1/ 5 { 1 + 1.8 T7T > (14) (Re Ra) Nu = 1.098 + 0.61 (Re Ra ) 1 / 5 (14a) Nu Nu o • 0.2189 (15) FIGURE 7 REPRODUCTION OF FIGURE 9 FROM REFERENCE 16 16 where Nu « 48/11 and x, i s given by o m 10? 6 + 5c 3 Pr - (25 Pr + 4)c 2 Pr + (20 Pr + I k Pr - 5 P r 2 = 0 m m m m Very recently Shannon and Depew [18] published an experimental i n -vestigation of flow through a horizontal c i r c u l a r tube with uniform wa l l heat f l u x . Water near the ice point was used as heat transfer medium. The Reynolds number was varied from 120 to 2300, and the Grashof number ranged up to 2.5 • 10 5 . No correlation formula was given, but they plotted t h e i r data as (Nu - Nu„ ) versus (Gr P r ) 1 / 4 / Nu„ as shown i n G Z G Z Fig. 8. Nu i s the Nusselt number measured i n t h e i r experiment, and Nu„ Gz i s a function of Graetz number evaluated from Siegel's [19] solution. Their difference represents the portion of Nusselt number which i s due to free convection. I t was f e l t that the natural convection i s dependent on the term (Gr Pr ) 1 / 1 * only, as suggested by Mikesell [21]. In the entrance region, the heat transfer rate depends on the ra t i o L/D, which can be taken into account by introducing the Graetz number. For the f u l l y developed flow region however, the heat transfer c o e f f i -cient w i l l be independent of the tube length, and hence the Nusselt num-ber w i l l not be a function of the Graetz number. Table 1 gives a summary of the region of dimensionless parameters covered by experimental data reported by references [14] - [18]. I t also contains the region covered by the present investigation. 1 1 1 I I 0 1 2 3. 4 5 (GrPr) 1 / 4 > N u G z FIGURE 8 REPRODUCTION OF FIGURE 6 FROM REFERENCE 18 Re Gr Pr Fluid Ede 300 - 100,000 Air / Water McComas & Eckert 100 900 1 - 1,000 0.75 A i r Mori 1,890 1,450 33 - . 5 7 0.75 A i r Shannon & Depew 120 2,300 up to 2.5.10 5 2 - .14 Water Present Investigation 100 2,000 300 - 30,000 4 - 9 Water TABLE 1 Summary of Experimental Investigations. 19 PURPOSL' OF EXPERIMENT As mentioned i n the l a s t chapter, the purpose of the present experi-ment was to obtain the heat transfer c o e f f i c i e n t , or Nusselt number, as a function of other dimensionless parameters. The apparatus was also de-signed to investigate the entrance length required for the flow to become fully-developed. DESCRIPTION OF EXPERIMENTAL EQUIPMENT Flow Loop A schematic diagram of the flow loop i s shown i n Fig. 9. The heat transfer medium used was d i s t i l l e d and deionized water. A constant temp-erature bath was used to maintain a steady i n l e t temperature. I t also provided a constant head of eight feet, assuring a constant flow rate. Water from the head tank flowed down the stand pipe, and temperature was measured before entering the test section. I t then passed through a smooth i n l e t to the hydrodynamic approach, then into the test section and to the out l e t . The i n l e t , tube, and outlet were placed i n a vacuum tube for heat i n s u l a t i o n . The hydrodynamic approach section has a length of one foot and an outside diameter of 0.25 inch. The length to diameter r a t i o of 52 allowed the ve l o c i t y p r o f i l e to become fully-developed. The s i x foot long test section was separated thermally from the hydrodynamic approach section by a tef l o n disk. The hydrodynamic ap-proach and test section are both from the same 10/1000 inch w a l l thick-COOLER II CONSTANT "BFERATURE BATH CARTRIDGES VACUUM PUMP Q VAcuuf-1 CHAMBER TEST SECTION FLOW f'ETER •cxi—©-COOLER I SUMP TANK CIRCULATING PUIT FIGURE 9 SCHEMATIC OF HEAT IRANSFER LOOP 21 ness Inconel tube. Thermocouples attached to the outer wall enabled de-termination of the l o c a l w a l l temperature. The t e s t section e x i t water temperature was measured before being cooled back to approximately room temperature. It then passed through a flowmeter and t h r o t t l i n g valve, where the flow rate could be adjusted, and into a sump tank.. The exact flow rate was measured by c o l l e c t i n g water over a known time i n t e r v a l . A pump c i r c u l a t e d the water e i t h e r through two cartridges i n s e r i e s , or d i r e c t l y i n t o a cooler. One of the cartridges contained a mixed bed ion-exchanger matrix to deionize the water, the other a matrix to remove or-ganic p a r t i c l e s from the water. Since the constant temperature bath con-tained only a heater, i n l e t temperature was maintained by f i r s t cooling the water i n a heat exchanger connected to a domestic cold water supply. F i g . 10 shows part of the experimental set up. Test Section The t e s t s e c t i o n was an Inconel tube with a 0.25 inch outside diame-ter and a w a l l thickness of 10/1000 of an inch. The length of 72 inches gave a length to diameter r a t i o of about 313. The outside diameter was checked for uniformity over the whole length. An o v e r a l l i n s i d e diameter was also measured, but the uniformity of the w a l l thickness was not checked. F i g . 11 shows d e t a i l s of the t e s t s e c t i o n . The t e s t s e c t i o n was desired to have high e l e c t r i c a l r e s i s t i v i t y , and low thermal conductivity. Hie former was to obtain a higher r e s i s t -ance and hence lower current by a higher voltage drop, the l a t t e r to mini-mize heat conduction along tube axis. A Nickel A l l o y used for resistance 22 T E S T SECTION 24 heating s a t i s f i e s t h i s condition best. A thi n tube wall i s used since t h i s w i l l reduce thermal conductance and raise the e l e c t r i c a l resistance. An "Inconel X 750" tube was found to be most suitable for t h i s . Thermocouples were mounted on the test tube with small non-conducting clamps i n order to maintain good contact between the tube wall and wire junction. To minimize error due to thermocouple conductance the thermo-couple wires were wound around the tube a few times. Due to secondary flow i n the test section, higher heat transfer rates are expected from the lower than the upper half of the tube c i r -cumference. This gives r i s e to non-uniform temperature over a circumfer-ence at any tube section. In an attempt to take t h i s into account, thermocouples were placed at three points at a tube section as shown i n Fig. 12. I Fig. 12: Thermocouple placing on test section circumference The a x i a l spacing of thermocouples was 12 inches i n the middle part of the tube, and less near each end i n order to allow a determination of the entrance length and to detect heat loss due to conductance along power leads and tube w a l l . The location of the thermocouples i s given 25 i n Appendix B. C i r c u l a r copper flanges were mounted at each end of the te s t s e c t i o n by a s l i g h t press f i t . The e l e c t r i c a l conductivity was im-proved by using conductive epoxy s i l v e r solder. The hydrodynamic approach section was thermally insulated from the te s t tube by a t e f l o n tube of one inch length. The i n l e t to the hydro-dynamic approach had a smooth converging shape, and was also made of t e f l o n . At the te s t section e x i t , a tube of 10 inches i n length of the same diameter was used before diverging to a larger diameter. In the larger s e c t i o n , a f i n e wire mesh was used to provide mixing. The t e s t s e c t i o n and hydrodynamic approach were placed on a red f i b r e material with low thermal conductivity. Test Section I n s u l a t i o n For the small heat t r a n s f e r c o e f f i c i e n t s encountered i n laminar flow, the i n s u l a t i o n of the t e s t section i s very important. Hallman [20] found that use of a vacuum chamber was f i v e times more e f f e c t i v e i n terms of i n s u l a t i o n and attainment of steady state conditions than ordinary insu-l a t i o n m aterial. Therefore t h i s device was applied i n t h i s experiment. A ten foot long s t e e l tube of four inches diameter was used as a vacuum chamber, i n which the te s t section and hydrodynamic approach were placed. 26 1 1 0 VOLT VOLTAGE & G U L A T O R UJJJJJJMi} /rrrrrrnt VAR iAC 'juuiuxwuub prrrrrr) TRANSFOWCR WATTMETER I T E S T S E C T I O N 1 FIGURE 1 3 SCHEMATIC OF ROWER SUPPLY 27 Power Supply The test section i s resistance heated by a power supply as shown i n Fig. 13. Alternating current was employed i n order to eliminate voltage pick up across the thermocouple junction. Current passes through a v o l t -age regulator with an accuracy of 1 per cent, and was adjusted by a Va-r i a c . A steady step-down transformer gave the required lower voltage. The current then passed through a wattmeter, and to power-leads attached to the test section. Temperature Measurement Copper-Constantan thermocouple wire of 30 gauge was used throughout for temperature measurement. There are two groups of thermocouples, as shown i n Fig. 14. The f i r s t group pertains to thermocouples where less accuracy i s required, for example room temperature, and a u x i l i a r y temper-atures of the flow loop such as temperature i n the cooler, flowmeter, etc. These temperatures were measured with a "Honeywell Electronic 15" i n s t r u -ment having an accuracy of 1°F. This instrument has 24 inputs, i s s e l f -balancing and does not need a reference junction. The second group of thermocouples measured temperature of the test section w a l l and the mix-ing cup temperature. They were selected by a multipole switch, measured either by the Honeywell instrument (for rough readings), or by the K3 Universal potentiometer (for accurate readings). The K3 instrument en-ables a voltage selection of 0.5 microvolt, and i s used with standard c e l l , galvanometer and reference junction. A melting ice bath was used for a reference junction. D i s t i l l e d water ice was used for t h i s purpose and was constantly s t i r r e d to maintain uniform temperature. 28 HONEYWELL-ELECTRONIC 1 5 SELECTOR SWITCH > PLEASURING > JUNCTIONS > COPPER CONSTANTAN MELTING ICE BATH G A L V A N O T C T E R FIGURE 1 4 THERMOCOUPLE CIRCUIT 29 EXPERIMENTAL PROCEDURE Test Procedure The experimental data were taken i n the following manner. A flow rate and a power l e v e l were set, and allowed to reach a steady value. When the i n l e t water temperature attained a steady value, readings were taken of the flow rate, power, water i n l e t , and o u t l e t temperature, both before and a f t e r reading the wall temperature thermocouples. Thermocouple C a l i b r a t i o n Method The t e s t section and mixing cup thermocouples were ca l i b r a t e d i n order to obtain better accuracy i n temperature measurements. These thermocouples were c a l i b r a t e d by comparing them with a standard mercury thermometer. For c a l i b r a t i o n , a l l thermocouples were detached from the test sec-t i o n and fastened to the bulb of the mercury thermometer. The hot junc-tions were then put i n t o a Dewar f l a s k together with the standard thermo-meter, and water of desired temperature placed i n the Dewar f l a s k . Each thermocouple voltage was read simultaneously on the potentiometer and mercury thermometer. This procedure was repeated for four d i f f e r e n t temperatures (roughly 170, 112, 89, and 55°F) covering the whole region i n which thermocouples were used for the test runs. During the c a l i b r a -t i o n the temperature of the i c e bath was also measured with a mercury thermometer, and found to vary by 0.1 centigrade. 30 Calibration Result The voltage reading of a l l thermocouples varied not more than f i v e microvolts;, which corresponds to 0.25°F. The accuracy of the mercury thermometer was 0.09°F. When the galvanometer was used, one could detect a voltage v a r i a t i o n not smaller than 3 to 4 microvolts. Since the ther-mocouples agreed with each other very c l o s e l y , they were not treated sep-arately but only one c a l i b r a t i o n curve was used for a l l thermocouples. The data from the c a l i b r a t i o n procedure were f i t t e d by method of least squares to a second degree polynomial equation of the form T = 32 + CXE + C 2E 2 (16) where T temperature, °F E microvolts C i , C 2 constants. The constants were evaluated, giving the following equation T = 32 + 4.763 • 10~2E - 1.507 • l O ^ E 2 . (16a) 31 RESULTS AND DISCUSSION Heat transfer data were obtained for a horizontal c i r c u l a r tube with i n t e r n a l laminar flow. The boundary condition was approximately uniform heat f l u x over the entire test section. The Reynolds number ranged from 100 to 2000. The Grashof number was varied by applying different rates of heat generation, and the natural convection effect on the forced lami-nar heat transfer was studied by comparing the high Grashof number runs to runs at lower Grashof number at approximately the same Reynolds num-ber. Since water was used as the heat transfer medium the Prandtl number could be changed only by using a dif f e r e n t temperature range. The flow was considered laminar when no wal l temperature f l u c t u -ations were present. While taking temperature readings, one could ob-serve when the flow was changing from steady laminar to random eddying flow. The l i m i t of laminar flow i s dependent on Reynolds number and Gras-hof number. I t was found that at a certain Reynolds number the t r a n s i - . t i o n flow was present i f heat f l u x was high enough, that i s , high Grashof number. A smaller Reynolds number s t a b i l i z e d the flow. I t was not the purpose of the present work to investigate the l i m i t s of laminar flow. These observations were merely to omit data i n the t r a n s i t i o n or turbu-lent flow region. 32 Entrance Length and Circumferential Temperature Variation Similar to the development of a laminar v e l o c i t y p r o f i l e , the tem-perature p r o f i l e needs a certain a x i a l distance to become f u l l y - d e v e l -oped. The entrance region i s determined from measurement of the a x i a l w a l l temperature gradient. Consider the f l u i d temperature p r o f i l e as shown i n Fig. 15. At the entrance to the heated section the tempera-ture i s uniform. As the f l u i d moves downstream, heat i s removed from the w a l l and the temperature difference between f l u i d and the w a l l i s i n -creased. In the tube center there i s a core of f l u i d which i s s t i l l at the same temperature as i n the unheated section. As the f l u i d moves further downstream the core diminishes and f i n a l l y disappears. This i s the point when the temperature p r o f i l e i s considered fully-developed. For pure forced convection and laminar flow, the entrance length w i l l be di f f e r e n t than the present case under investigation. Fi g . 16 shows temperature p r o f i l e s for run number 42. Between X/D = 100 and 300 the Reynolds number increases from 669 to 742, and the Gras-hof number from 1269 to 1541. The flow can be considered fully-developed approximately from the point X/D =75. At X/D = 104 the difference be-tween bottom and top wa l l temperature i s 2.6°F, while further downstream the difference between bottom and top w a l l temperature increases to be-come a maximum of 5.9°F at about X/D.= 200, and then decreases s l i g h t l y towards the test section end. I t may be noted from Fig. 16 that i n the entrance region the wa l l temperature gradient i s high at the tube i n l e t , and decreases slowly i n LU CC Z3 LU s 1 0 0 9 5 9 0 8 0 7 5 7 0 5 0 X/D 1 0 0 3 0 0 RE 1 2 9 . 1 4 6 PR 8,4 7,2 GR 3 0 9 5 4 9 RA 2 5 9 2 3 9 6 3 o AVERAGE WALL TEMPERATURE o TOP WALL TEMPERATURE o S I D E WALL TEMPERATURE v BOTTOM WALL TEMPERATURE JL JL 0 1 0 0 2 0 0 A X I A L DISTANCE- X/D 3 0 0 FIGURE 1 5 REPRESENTATIVETEMPERATURE P R O F I L E RUN NUMBER 3 4 0 0 1 0 0 9 0 o L U or c. X/D 1 0 0 3 0 0 RE 6 6 8 742 PR 7.8 7.1 GR 1 2 6 9 1 5 5 3 RA 9 8 7 1 1 1 0 2 9 8 0 o AVERAGE WALL TEMPERATURE n TOP WALL TEMPERATURE • S I D E WALL TEMPERATURE v BOTTOM WALL TEMPERATURE 5 0 1 0 1 0 0 2 0 0 A X I A L DISTANCE X/D 3 0 0 4 0 0 FIGURE 1 6 REPRESENTATIVE TEMPERATURE P R O F I L E RUN NUMBER 4 2 35 the a x i a l d i r e c t i o n . At about 100 diameters downstream i t becomes almost constant. A x i a l Temperature P r o f i l e s Fig. 17 shows the tube wall temperature and the f l u i d bulk tempera-ture for run number 1. Between X/D = 100 and 300 the Reynolds number varies from 366 to 408, and the Grashof number from 703 to 965. Down-stream of X/D = 100 the average w a l l temperature remains constant. The difference between f l u i d bulk and mean w a l l temperature i s 5.7°F at X/D » 100, and decreases s l i g h t l y to 4.6°F at the test section e x i t . The difference between bottom and top wal l temperature i s approximately 1.5°F, or nearly 30 per cent of the difference between mean wall and f l u i d bulk temperature. Fig. 18 shows the temperature p r o f i l e s for run number 43. The Rey-nolds number i s 331 at the i n l e t to the heated section. This test run was performed with higher heat f l u x , giving a Grashof number of the order of 2000. The gradient of the average wa l l temperature decreases u n t i l approximately X/D =60; from here on downstream i t remains constant. Average w a l l and f l u i d bulk temperature p r o f i l e s are not p a r a l l e l . The difference between f l u i d bulk and mean wall temperature i s 8.8°F at X/D = 100, and decreases to 6.7°F at X/D = 300. One should not expect the f l u i d bulk temperature gradient and the mean wal l temperature gradient to be exactly the same. With change of the f l u i d bulk temperature the properties of water change, and this cau-ses a change i n the dimensionless numbers such as Reynolds, Grashof, and 1 0 0 i i i r ~ — r 9 0 X/D 1 0 0 3 0 0 RE 3 6 6 4 0 9 PR '8.2 7.2 GR 7 0 2 9 6 5 RA 5 7 6 6 6 9 7 5 o LU 8 0 o AVERAGE WALL TEMPERATURE n TOP WALL TEMPERATURE o S I D E WALL TEMPERATURE v BOTTOM WALL TEMPERATURE LU LU FIGURE 1 7 1 0 0 2 0 0 A X I A L DISTANCE X/D REPRESENTATIVE TEMPERATURE P R O F I L E RUN NUMBER 1 3 0 0 4 0 0 37 1 0 0 LU 3 UJ 9 0 8 0 X/D 1 0 0 3 0 0 RE 3 6 4 4 4 0 PR 7.4 6,0 GR 1 5 7 1 , 2 5 2 3 RA 1 1 6 2 8 1 5 0 7 6 7 0 6 0 o AVERAGE WALL TEMPERATURE a TOP WALL TEMPERATURE ^ S I D E WALL TEMPERATURE v BOTTOM WALL TEMPERATURE 5 0 1 0 1 0 0 2 0 0 A X I A L DISTANCE X/D FIGURE 1 8 REPRESENTATIVE TEMPERATURE P R O F I L E .«RUN NUMBER 4 3 3 0 0 4 0 0 38 Prandtl numbers. The heat f l u x i s the same over the entire te.^t section, but a change i n the free convection effect influences the heat transfer rate, therefore the temperature difference w i l l change. In the test run considered, (Fig. 18), the Reynolds number increases from 364 at X/D = 104, to 440 at X/D = 306. In the same section the Prandtl number changes from 7.4 to 6.0, and the Grashof number increases from 1571 to 2523. Fig . 19 shows the temperature p r o f i l e of run number 37, which cov-ered Reynolds numbers In the region of 1150 to 1300. The Grashof number varied approximately from 2300 to 3600. The temperature difference i n the entrance region increased r a p i d l y , then decreased s l i g h t l y towards the test section e x i t . Fig. 20 shows the temperature p r o f i l e for a test run with high flow rate *and heat f l u x . The Reynolds number varies from 1374 to 1600, and the Grashof number from 6000 to 9600. The temperature difference between mean w a l l and f l u i d bulk i s nearly 20°F at X/D = 200. A large circumfer-e n t i a l temperature difference i s produced due to the high heat f l u x . When Figs- 17, 18, 19, and 20 are compared with each other, one no-t i c e s a consistent droping of the average wa l l temperature near the tube e x i t . This droping i s more pronounced i n higher Grashof number runs. -Local Nusselt Number Fi g . 21 shows the Nusselt number p r o f i l e for a run with Reynolds numbers i n the order of 130, and with Grashof numbers ranging from 117 to 500. The curve closely resembles the prediction for pure forced-convec-FIGURE 1 9 REPRESENTATIVE TEMPERATURE P R O F I L E , RUN NUMBER 3 7 1 1 0 1 1 1 X/D 1 0 0 3 0 0 - RE 1 3 7 4 1 6 1 1 PR 6,1 5,1 GR 7 1 2 9 1 0 4 2 1 " RA 4 3 4 7 0 5 3 0 3 9 1 0 0 o AVERAGE WALL TEMPERATURE D TOP WALL TEMPERATURE o S I D E WALL TEMPERATURE v BOTTOM WALL TEMPERATURE 0 1 0 0 2 0 0 3 0 0 4( A X I A L DISTANCE X/D FIGURE 2 0 REPRESENTATIVE TEMPERATURE P R O F I L E RUN NUMBER 2 7 I I I I I I I 0 DO 200 300 CO AXM. DISTANCE X/D FIGURE 21 frmaumrm NUWELT NUPHER PROFILE RJN fern* 3 42 tior (dashed l i n e ) . The Nusselt numbers are slightly higher than 4.36, the value predicted for pure forced convection. Fig. 22 represents a Nusselt number profile with the Reynolds number in the order of 1100 to 1300 and the Grashof number of about 20,000. Due to the high Reynolds number, the temperature profile requires a larger distance to become established inspite of the very high Grashof number. Fig. 23 shows the plot of Nusselt number versus axial tube distance for run number 1. Reynolds numbers are of the order of 370, while Gras-hof numbers vary from 700 to 970. At the beginning of the heated section there i s a high heat transfer rate. The Nusselt number then decreases rapidly as the temperature profile develops, and at X/D = 100 the curve reaches its. minimum. This point coincides with the temperature profile becoming fully-developed (Fig. 23). From this point onwards the Nusselt number increases with approximately constant slope. Fig. 24 represents the Nusselt number variation for run 43. Rey-nolds and Grashof numbers are in the order of 370 and 1600 respectively. At the tube inlet the Nusselt number is approximately 9. It decreases to a minimum at X/D = 60, and then increases at a constant rate. Comparing run number 1 and 43 (replotted in Fig. 25), we notice that in the second case flow development is achieved in shorter distance, that i s a higher Grashof number seems to accelerate flow development. At X/D «* 104 i t appears that the Reynolds numbers are the same. It is interesting to study the effect of Grashof number on Nusselt number at this point. It appears that the higher Grashof number causes an increase in the 43 12 ID 8 U 6 h 2 I-0 AXIAL DISTANCE X/D FIGURE 22 REWESOITATI* Hussar NUMBER PROFILE RUN NUMBER 37 0 100 200 3 0 0 4 0 0 A X I A L DISTANCE X/D FIGURE 23 REPRESENTATIVE NUSSELT NUMBER P R O F I L E RUN NUMBER 1 12 1 0 45 cc L U _l L U C O C O r> 8 =3 6 X/D 1 0 0 3 0 0 RE 3 6 4 4 4 0 PR 7.4 6.0 GR 1 5 7 1 2 5 2 3 RA 1 1 6 2 8 1 5 0 7 6 I i i 2 0 0 3 0 0 0 1 0 0 4 0 0 AXIAL DISTANCE X/D FIGURE 2 * REPRESENTATIVE NUSSELT NUMBER PROFILE RUN SLUMBER 43 22 JD 8 ^ ^ • X ' " p ° — RUN NURSES 43 V RUN NUMBER 1 ° 0 100 200 300 400 AXIAL DISTANCE X/D FIGURE 25 tewestNTATivE NUSSELT NUMBER PROFILES RUN NUMBERS 1 AND 43 47 Nusselt number from 5.2 i n run 1 to 5.6 i n run 43. Furthermore the Nusselt number increases at a greater rate. heat Transfer Correlation Correlation of o v e r a l l heat transfer c o e f f i c i e n t s for the entire test section length with combined free and forced convection under uni-form heat fl u x i s complicated by the fact that the free convection effects do not st a r t at the tube i n l e t , but require a s t a r t i n g length to be established. For uniform w a l l temperature, both the r a t i o D/L and L/D were used [8] to give f a i r correlation. For uniform w a l l heat flux i t appears that there i s a pronounced development region to establish the secondary flow, and a region of thermally developed flow where- the heat transfer c o e f f i c i e n t s are dependent only on the parameters Gr, Re, Pr. Several correlations were tested on the data. The expression Nu = Y| + 0.047 P r 1 / 3 (Re R a) 1/ 5 (17) appears to correlate 53 per cent of the data (plotted i n Fig. 26) i n the fully-developed region to within ± 10 per cent. Another s l i g h t l y more accurate expression which correlates 68 per cent of the data to within ± 10 per cent but does not s a t i s f y the pure forced convection case i s Nu = 2.41 + 0.082 P r 1 / 3 (Re R a ) 1 / 5 (18) This data i s plotted i n Fig. 27. A point of caution might be added here. The above correlations do not apply to f l u i d s of Prandtl numbers higher than 10, where the additional factor of v a r i a t i o n of v i s c o s i t y with 48 7 0 6 0 5 0 U or O 4 0 h -3 0 H i 3 2 0 1 0 U 0 (RE RA)1/5 FIGURE 26 CORRELATION OF NUSSELT NUMBER, EQUATION 1 7 49 7 0 6 0 5 0 Q_ 9 OJ i 3 3 0 K 2 0 h io r 0 l l l l — o o o — o °<> • o w _ Q °8oo ° o oo °o° *<> °o o °A«o ° o o o _ ^ — oo° V 0 0 0 — ° o 0 o© o 0 ° - — l l l l 1 0 2 0 3 0 4 0 5 0 (RE RA)1/5 FIGURE 2 7 CORRELATION OF NUSSELT NUMBER, EQUATION 1 8 50 temperature i s dominant also. The expressions (17) and (18) should be considered as very tentative. Further studies using various other f l u i d s , tube s i z e s , and lengths are required i n order to v e r i f y these correlations, and perhaps shed further l i g h t on the basic nature of t h i s problem. M. 51 CONCLUSION From t h i s investigation experimental data was obtained for com-bined free and forced convection laminar heat transfer for water flowing through a c i r c u l a r horizontal tube with uniform heat f l u x . Results show that at low values of Grashof number the Nusselt number i s i n good agree-ment with t h e o r e t i c a l prediction for pure forced convection. High Gras-hof numbers reveal that the free convection effect w i l l create substan-t i a l increase i n Nusselt number. Two tentative correlations for the Nusselt number for fully-developed flow are given i n equation (17) and (18) as a function of Re Ra and Pr. 52 REFERENCES 1. A..°. Colburn, "A Method of Correlating Forced Convection Heat Transfer Data and a Comparison with Fluid Friction," Trans. AIChE, vol. 29, 1933, pp. 174 - 210. 2. E.N. Sieder and G.E. Tate, "Heat Transfer and Pressure Drop of L i -quids in Tubes," Ind. Eng. Chem., vol. 28, 1936, pp. 1429 - 1435. 3. D.G. Kern and D.F. Othmer, "Effect of Free Convection on Viscous Heat Transfer in Horizontal Tubes," Trans. AIChE, vol. 39, 1943, pp. 517 - 555. 4. R.C. Martinelli, C.J. Southwell, G. Alves, H.L. Craig, E.B.  Weinberg, N.F. Lansing., and. L.M.K. Boelter, "Heat Transfer and Pressure Drop for a Fluid Flowing in the Viscous Region through a Vertical Pipe," Heat Transfer Data, Part I, Trans. AIChE, vol. 38, 1942, pp. 493 - 530. 5. O.C. Eubank and M.S. Proctor, "Effect of Natural Convection on Heat Transfer with Laminar Flow in.Tubes," M.S. Thesis in Chemical Engineering, Massachusetts Institute of Technology, Cambridge, 1951. 6. W.H. McAdams, "Heat Transmission," Third Edition McGraw H i l l Book Co.,Inc., New York, 1954, p. 235. 7. T.W. Jackson, J.M. Spurlock, and K.R. Purdy, "Combined Free and Forced Convection in a Constant Temperature Horizontal Tube," Journal AIChE, vol. 7, 1961, pp. 38 - 41. 53 8. D.R. Oliver, "The Effect of Natural Convection on Viscous Flow Heat Transfer in Horizontal Tubes," Chemical Engineering Science, vol. 17, 1962, pp. 335 - 350. 9. A.R. Brown and M.A. Thomas, "Combined Free and Forced Convection Heat Transfer for Laminar Flow in Horizontal Tubes," Journal Mechan-i c a l Engineering Science, vol. 7, 1965, pp. 440 - 448. 10. J.J. Martin and M.P. Carmichael, "Combined Forced and Free Convec-tive Heat Transfer in a Horizontal Pipe," ASME, Paper No. 55-A-30, 1955. 11. B.R. Morton, "Laminar Convection in Uniformly Heated Horizontal Pipes at Low Rayleigh Numbers," Quarterly Journal of Mechanics and Applied Mathematics, vol. 12, no. 4, 1959, pp. 410 - 420. 12. M. Iqbal and J.W. Stachiewicz, "Influence of Tube Orientation on Combined Free and Forced Laminar Convection Heat Transfer," Journal of Heat Transfer, Trans. ASME, Series C, 1966, pp. 109 - 116. 13. E. Del Casal and N.N. G i l l , "A Note on Natural Convection Effects in Fully Developed Flow," Journal AIChE, vol. 8, 1962, pp. 570 - 574. 14. A.J. Ede, "The Heat Transfer Coefficient for Flow in a Pipe," International Journal of Heat and Mass Transfer, vol. 4, 1961, pp. 105 - 110. 15. S.T. McComas and E.R.G. Eckert, "Combined Free and Forced Convec-tion in a Horizontal Circular Tube," Journal of Heat Transfer, Trans. ASME, Series C, 1966, pp. 147 - 153. 54 16. Y. Mori, K. Futagami, S. Tok.ida, and M. Nakamura, "Forced Convec-tive Heat Transfer in Uniformly Heated Horizontal Tubes," Inter-national Journal of Heat and Mass Transfer, vol. 9, 1966, pp. 453 -463. 17. Y. Mori and K. Futagami, "Forced Convective Heat Transfer in Uni-formly Heated Horizontal Tubes," International Journal of Heat and Mass Transfer, vol. 10, 1967, pp. 1801 - 1813. 18. R.L. Shannon and C.A. Depew, "Combined Free and Forced Laminar Convection in a Horiz ontal Tube with Uniform Heat Flux," ASME Paper No.. 67-HT-52, 1967. 19. R. Siegel, E.M. Sparrow, and T.M. Hallman, "Steady Laminar Heat Transfer in a Circular Tube with Prescribed Wall Heat Flux," Applied Scienti f i c Research, Section A, vol. 7, 1958, p. 386. 20. T.M. Hallman, "Combined Forced and Free Convection in a Vertical Tube," PhD Dissertation, Purdue University, West Lafayette, Indiana, 1958. 21. R.D. Mikesell, "The Effect of Heat Transfer on the Flow in a Hori-zontal Pipe," PhD Thesis, Chemical Engineering Department, University of I l l i n o i s , 1963. 22. Anon., "Resistance Welding of Nickel and High-Nickel Alloys," Technical Bulletin T-33, The International Nickel Company. APPENDIX A DERIVATION OF GOVERNING EQUATIONS Equations (1) to (3) were derived from continuity, momentum and energy equations i n c y l i n d r i c a l coordinates. Flow i s considered f u l l y -developed, and physi c a l properties of the f l u i d are assumed constant ex-cept the density v a r i a t i o n a f f e c t i n g the buoyancy terms. In the energy equation a x i a l conduction, viscous d i s s i p a t i o n and pressure terms are ignored. The equations are then further reduced with help of the stream function \\i and expressed as rv r _ _aj>_ v 96 v _e = _ i l v 9r and the parameters v T - T R = — : V = -2- ; T* = W a ' x v/a ' A a Pr With these parameters the dimensionless numbers become Re = -Ca 3/(4pv 2) Gr = BgAaVv 2 Pr = c p/k P Ra = Gr Pr APPENDIX B LOCATION OF TEST SECTION THERMOCOUPLES Thermo-couple Number Distance from Test Section I n l e t [in] Distance i n Diam-eters Location on circum-ference 57 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 1 1 2 3 4 5 6 7 8 9 10 11 12 12 12 24 24 24 36 4 4 8 13 17 22 26 30 35 39 43 48 52 52 52 104 104 104 157 top bottom top top top top top top top top top top top side bottom top side bottom top Thermo-couple Number Distance from Test Section Inlet [in] Distance i n Diam-eters Location on circum-ference 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 36 36 48 48 48 60 60 60 66 68 20 70.5 70.5 70.5 16 157 157 209 209 209 261 261 261 287 296 87 307 307 307 70 side bottom top side bottom top bottom side top top top top side bottom top 58 APPENDIX C SAMPLE CALCULATION The data of run number 3 are chosen for purposes of il l u s t r a t i o n . The measured quantities are Flow rate 41.6 ccm/min at 15°C Power input 25.0 watts Temperature distribution as given in Fig. 15 (p.33) The heat flux was calculated from the bulk temperature rise and flow rate. The temperatures given in Fig. 15, were obtained using the thermocouple calibration results. A mean outside wall temperature was calculated from the three thermocouples placed at different circumferential locations (Fig. 12, p.24). The mean wall temperature i s faired by a straight line, and shows that maximum deviation i s very small. The bulk temperature was measured before the test section inlet and after the test section exit, and i t s gradient assumed to be constant since the uniformity of heat generation over the tube length could not be checked. The flow was considered fully-developed away from the region of steep temperature gradient. For this run, this occurred approximately at X/D =75. At the location X/D = 104, the recorded mean wall and bulk 59 temperature were T - 59.8°F T = 57.0°F The heat transfer c o e f f i c i e n t i s based on inner wall temperature, and the temperature drop through the wal l was calculated from * - 2ir { - R 2 l n 7 } ( ( > 1 ) w where W heat generation rate per unit volume of tube material k w thermal conductivity of wal l ( k w =8.67 BTU/hr °F f t [22]) R outside radius = 0.125 inch r inside radius = 0.115 inch For t h i s apparatus (C-l) reduces to T = 4.19 10" 4 P (C-2) where P i s test section power input i n watts. Equation (C-l) was derived with the assumption of uniform power density i n the tube wa l l and no heat loss at the outer surface. Furthermore, a uni-form a x i a l temperature gradient i n the tube wall and constant physical properties of the tube were assumed. The maximum calculated temperature drop across the tube w a l l was 0.24°F, and for t h i s run was about 0.01°F. 60 Heat Transfer C o e f f i c i e n t and Nusselt Number The Nusselt number i s defined as Nu = (C-3) fcf where h stands f o r the heat t r a n s f e r c o e f f i c e n t , h = Y^IT (C-4) w and q i s the heat flow rate. The thermal conductivity of water was based on the bulk temperature. The heat f l u x was calculated from the flow rate and the bulk temperature r i s e where q = m Cp A/area m mass flow rate Cp s p e c i f i c heat of bulk temperature A bulk temperature gradient Area surface area per foot length. Therefore the heat f l u x and Nusselt number are q = 0.061 BTU/ft 2sec Nu = 4.52 61 Reynolds Number The Reynolds number is defined as uD m 4 Re = — or — v p TTVD i f the mean velocity u i s expressed in terms of the flow rate so that for this case Re = 129 . ' > . The viscosity was evaluated at water bulk temperature. Grashof Number The Grashof number is here defined as Gr 3 g AT D3 (C-6) where 3 coefficient of thermal expansion g gravitational acceleration AT temperature difference beteen bulk fl u i d and inner wall, as calculated For this point c Gr = 309 62 APPENDIX D ERROR ESTIMATES To give some idea of the accuracy of the data, error estimates were performed. The accuracy varied a great deal between runs and for various positions on the test section. Because of the number of variables which affect the r e s u l t s , i t i s impractical to present estimated errors for each data point, and error estimates for only two cases are presented below. Error i n mass flow rate 0.5% < m < 1.5% Possible error i n water temperature at i n l e t to test section 0.25°F < T < 0.5°F Possible error i n water temperature at e x i t of test section 0.5°F < T < 1.0°F Error i n tube wa l l temperature measurements for heated section Tw = 0.25°F The errors i n water temperature are noticed by s l i g h t f l u c t u a t i o n of the measured temperature. They are due to the error i n temperature measure-ments and due to the inherent i n s t a b i l i t y of the flow system as w e l l as the power generation. Possible errors i n property values such as thermal 63 conductivity and v i s c o s i t y due to error i n temperature may be as much as 0.2%. 1 Probable Errors for Results For run number 3 Difference from wa l l to bulk temperature AT = 2.75°F A x i a l temperature gradient of f l u i d bulk A = 2.42°F/ft From error i n w a l l and bulk temperature the bulk temperature gradient may be i n error of 3.4% Error i n temperature difference T = 0.5°F -• 15.2% Probable error of dimensionless numbers are Re = ~ r = 2.0% pvD G r = B R AT 18.5% Error estimated for run number 27 At location X/D = 200 the temperature difference i s T = 19.7°F and the bulk temperature gradients i s A = 3.54°F/ft 64 Error of dif f e r e n t components are mass flow rate 0.5% wa l l temperature 0.25°F f l u i d bulk temperature at i n l e t 0.25°F f l u i d bulk temperature at e x i t 1.0°F temperature difference 1.25°F = 6.35% bulk temperature gradient = 4.7% And the error of dimensionless numbers are Nusselt number Reynolds number Grashof number 7.95% 1.8% 6.35% 65 APPENDIX E TABULATED LOCAL HEAT TRANSFER DATA Run Number 1, Q = 0.146 BTU/sft sec X Nu Re Pr Gr Ra 1 8.6 347 8.8 316 2767 3 7.4 350 8.7 384 3339 5 6.3 352 8.6 465 4021 7 6.0 349 8.7 460 4017 9 5.9 356 8.5 526 4483 12 6.0 357 8.5 527 4473 24 5.2 366 8.2 702 5766 36 5.5 376 7.9 770 6108 48 5.6 387 7.7 850 6524 60 5.9 399 7.4 915 6794 70.5 6.2 409 7.2 965 6975 66 TABULATED LOCAL HMT TRANSFER DATA Run Number 2, Q = 0.121 BTU/sft. sec X Nu Re Pr Gr Ra 1 8.1 124 8.8 275 2413 3 6.3 126 8.7 381 3304 5 5.5 127 8.6 457 3907 7 5.3 129 8.4 506 4277 9 4.9 130 8.3 583 4858 12 4.9 132 8.2 634 5187 24 4.8 141 7.6 867 6563 36 4.8 151 7.0 1146 8050 48 4.7 161 6.5 1470 9593 60 4.7 171 6.1 1847 11214 70.5 4.7 180 5.7 2203 12637 67 TABULATED LOCAL HEAT TRANSFER DATA Run Number 3, Q = 0.061 BTU/sft sec X Nu Re Pr Gr Ra 1 8.1 120 9.1 117 1061 3 6.2 121 9.0 159 1429 5 5.0 122 8.9 206 1840 7 4.8 123 8.9 225 1987 9 4.8 124 8.8 235 2063 12 4.8 125 8.7 248 2154 24 4.5 129 8.4 309 2592 36 4.4 133 8.0 378 3039 48 4.6 137 7.7 423 3277 60 4.6 142 7.5 488 3637 70.5 4.5 146 7.2 549 3963 68 TABULATED LOCAL HEAT TRANSFER DATA Run Number 11, Q = 0.394 BTU/sft sec X Nu Re Pr Gr Ra 1 9.8 534 8.4 943 7873 3 7.2 538 8.3 1336 11049 5 6.4 543 8.2 1578 12906 7 6,1 547 8.1 1724 13955 9 5.9 552 8.0 1850 14831 12 5.4 559 7.9 2135 16844 24 5.7 590 7.4 2542 18882 36 6.3 621 7.0 2861 20030 48 6.9 654 .6.6 3155 20837 60 7.8 687 6.2 3358 20939 70.5 8.6 716 5.9 3478 20680 69 TABULATED LOCAL HE4.T TRANSFER DATA Run Number 12, Q » 0.527 BTU/si: sec X Nu Re Pr Gr Ra 1 10.4 540 8.3 1209 10032 3 8.2 546 8.2 1627 13323 5 7.4 552 8.1 1916 15463 7 6.6 558 8.0 2274 18108 9 6.3 565 7.9 2498 19617 12 6.3 571 7.8 2644 20503 24 6.3 616 7.1 3629 25805 36 6.9 659 6.6 4301 28299 48 7.6 705 6.1 4884 29781 60 8.6 750 5.7 5397 30659 70.5 9.6 788 5.4 5688 30566 TABULATED LOCAL Run Number 13, Q = 0.479 X Nu Re 1 13.1 1008 3 9.3 1015 5 8.3 1020 7 7.7 1025 9 7.3 1031 12 7.0 1038 24 6.2 1070 36 5.8 1104 48 5.7 1141 60 5.7 1179 70.5 5.7 1212 HEAT TRANSFER DATA BTU/sft sec Pr Gr Ra 8.7 695 6064 8.7 1017 8815 8.6 1169 10062 8.6 1296 11092 8.5 1409 11982 8.4 1550 13051 8.1 2046 16582 7.8 2547 19884 7.5 2959 22251 7.2 3431 24859 7.0 3833 26936 71 TABULATED LOCAL HEAT TRANSFER DATA Run Number 14, Q = 0.474 BTU/sft sec X Nu Re Pr Gr Ra 1 15.9 1000 8.8 540 4758 3 9.9 1007 8.7 899 7863 5 8.6 1014 8.7 1074 9329 7 7.9 1021 8.6 1209 10425 9 7.5 1026 8.6 1307 11199 12 7.4 1032 8.5 1378 11707 24 6.7 1063 8.1 1788 14629 36 6.2 1097 7.9 2262 17823 48 5.7 1133 7.6 2818 21398 60 5.3 1171 7.3 3472 25405 70. 5 5.1 1202 7.1 4083 28995 TABULATED LOCAL HEAT TRANSFER DATA Run Number 17, Q = 0.502 BTU/sft sec X Nu Re Pr Gr Ra 1 16.1 1425 8.9 546 4846 3 12.4 1430 8.8 721 6375 5 10.7 1433 8.8 842 7426 7 8.4 1448 8.7 1138 9937 9 8.0 1455 8.7 1226 10648 12 7.5 1465 8.6 1348 11625 24 6.8 1496 8.4 1696 14224 36 6.3 1530 8.2 2066 16846 48 5.8 1565 7.9 2489 19732 60 5.4 1602 7.7 2964 22873 70.5 5.2 1634 7.5 3408 25709 73 TABULATED LOCAL Run Number 27, Q = 0.720 X Nu Re 1 12.2 1259 3 9.6 1269 5 8.7 1278 12 8.0 1312 24 7.2 1374 36 7.3 1437 48 7.2 1497 60 8.0 1559 70.5 8.5 1611 HEAT TRANSFER DATA BTU/sft sec Pr Gr Ra 6.7 3057 20618 6.7 4005 26773 6.6 4567 30254 6.4 5465 35117 6.1 7129 43470 5.8 8134 47133 5.5 9480 52503 5.3 9699 51261 5.1 10421 53039 74 TABULATED LOCAL Hi'AT TRANSFER DATA Run Number 29, Q = 0.996 BTU/sfr. sec X Nu Re Pr Gr Ra 1 15.5 1102 8.1 1734 13989 3 10.8 1114 8.0 2625 20900 5 9.5 1126 7.9 3116 24504 7 8.9 1138 7.8 3525 27381 9 8.5 1151 7.7 3853 29553 12 8.2 1171 7.5 4309 32418 24 7.7 1252 7.0 5960 41601 36 7.9 1332 6.5 7353 47765 48 8.1 1417 6.1 8896 53839 60 8.3 1501 5.7 10545 59789 70.5 8.5 1571 5.4 11954 64383 75 TABULATED LOCAL HEAT TRANSFER DATA Run Number 31, Q = 0.754 BTU/sft sec X Nu Re Pr Gr Ra 1 15.3 1583 8.2 1223 10069 3 12.0 1591 8.2 1608 13147 5 9.7 1599 8.1 2039 16559 7 9.0 1608 8.1 2254 18188 9 8.7 1616 8.0 2385 19112 12 8.5 1631 7.9 2569 20357 24 8.8 1689 7.6 2876 21915 36 8.8 1749 7.3 3367 24653 48 8.4 2037 6.1 6166 37935 60 8.5 1872 6.8 4524 30692 70.5 8.0 1926 6.6 5328 34981 TABULATED LOCAL HEAT TRANSFER DATA Run Number 32, Q = 0.989 BTU/sft sec X Nu Re Pr Re 1 11.5 1600 8.2 2218 3 10.8 1606 8.1 2406 5 9.5 1614 8.1 2816 7 8.8 1625 8.0 3119 9 8.5 1634 8.0 3329 12 8.8 1658 7.9 3437 24 8.0 1734 7.4 4581 36 9.9 1815 7.0 4445 48 9.0 1903 6.7 4395 60 8.2 1977 6.4 7353 70.5 8.1 2045 6.1 8383 77 TABULATED LOCAL HEAT TRANSFER DATA Run Number 34, Q - 0.254 BTU/sft sec X Nu Re Pr Gr Ra 1 9.0 1577 8.2 699 5755 3 8.6 1581 8.2 746 6121 5 8.3 1585 8.2 782 6391 7 8.2 1587 8.2 800 6528 9 7.9 1589 8.1 831 6769 12 7.9 1592 8.1 839 6819 24 7.2 1609 8.0 970 7777 36 6.6 1628 7.9 1125 8903 48 6.3 1646 7.8 1233 9626 60 6.5 1666 7.7 1268 9772 70.5 6.9 1685 7.6 1252 9523 78 TABULATED LOCAL HEAT TRANSFER DATA Run Number 35, Q = 0.231 BTU/sft sec X Nu Re 1 10.8 1106 3 9.7 1109 5 9.1 1112 7 8.6 1115 9 8.2 1117 12 7.9 1120 24 6.6 1137 36 5.9 1155 48 5.7 1173 60 5.8 1191 70.5 6.4 1210 Pr Or Ra 8.1 564 4570 8.1 637 5148 8.1 686 5523 8.0 736 5905 8.0 786 6294 8.0 825 6580 7.8 1064 8341 7.7 1268 9772 7.6 1409 10671 7.4 1471 10946 7.3 1415 10342 79 TABULATED LOCAL HEAT TRANSFER DATA Run Number 36, Q = 0.258 BTU/sft sec X Nu Re Pr Gr Ra 1 11.1 1073 8.3 546 4551 3 10.0 1075 8.3 612 5085 5 9.3 1078 8.3 668 5533 7 9.0 1081 8.3 701 5791 9 8.4 1083 8.2 759 6255 12 8.3 1088 8.2 789 6458 24 7.1 1107 8.0 1012 8111 36 6.4 1124 7.9 1212 9530 48 6.3 1143 7.7 1323 10215 60 6.7 1164 7.6 1349 10201 70.5 7.3 1181 7.4 1307 9724 80 TABULATED LOCAL HEAT TRANSFER DATA Run Number 37, Q =0.500 BTU/sfL sec X Nu Re Pr Gr Ra 1 12.4 1085 8.3 956 7946 3 9.4 1088 8.3 1277 10577 5 8.9 1090 8.3 1379 11385 7 8.0 1093 8.2 1556 12809 12 7.8 1116 8.0 1766 14155 24 6.6 1156 7.7 2456 18895 36 6.5 1195 7.4 2850 21130 48 6.2 1235 7.1 3462 24734 60 7.4 1275 6.9 3284 22635 70.5 7.6 1311 6.7 3579 23891 81 TABULATED LOCAL HEAT TRANSFER DATA Run Number 38, Q = 0.497 BTU/sft sec X Nu Re Pr Gr Ra 1 11.3 656 8.3 1049 8720 3 8.6 660 8.2 1445 11873 5 8.2 656 8.1 1571 12780 7 7.5 672 8.0 179 14441 9 7.1 678 8.0 1994 15855 12 6.9 687 7.8 2186 17114 24 6.9 726 7.4 2784 20480 36 6.8 757 7.0 3317 23258 48 7.6 807 6.5 3768 24585 60 8.0 849 6.1 4262 26220 70.5 8.4 885 5.9 4678 27464 82 TABULATED LOCAL HEAT TRANSFER DATA Run Number 39, Q = 0.972 BTU/sft sec X Nu Re Pr Gr Ra 1 9.9 * 662 8.2 2427 20012 3 8.5 673 8.1 3094 24961 5 7.9 685 7.9 3603 28471 7 7.6 696 7.8 4042 31351 9 7.3 709 7.6 4540 34484 12 7.7 728 7.4 4817 35486 24 8.0 808 6.5 6973 45647 36 8.2 893 5.8 9517 55567 48 8.6 976 5.3 12221 64500 60 9.0 1065 4.8 15282 73061 70.5 9.3 1138 4.4 18064 80018 TABULATED LOCAL HEAT TRANSFER DATA Run Number 43, Q = 0.234 BTU/sft sec X Nu Re Pr Gr Ra 1 8.0 331 8.3 703 5826 3 6.5 334 8.2 906 7434 5 5.9 336 8.1 1035 8406 7 5.6 339 8.0 1145 9212 9 5.5 342 8.0 1207 9609 12 5.4 346 7.8 1304 10232 24 5.6 364 7.4 1571 11628 36 5.9 383 7.0 1829 12787 48 6.2 403 6.6 2086 13789 60 6.6 422 6.3 2340 14630 70.5 7.0 440 6,0 2523 15076 84 TABULATED LOCAL Run Number 44, Q = 0.440 X Nu Re 1 9.8 335 3 7.7 340 5 6.6 344 7 6.3 349 9 6.0 354 12 6.2 366 24 6.4 401 36 6.5 439 48 6.6 477 60 6.7 516 70.5 6.9 550 HEAT TRANSFER DATA TU/sft sec Pr Gr Ra 8.2 1143 9363 8.1 1558 12548 7.9 1923 15263 7.8 2140 16728 7.7 2376 18281 7.4 2664 19733 6.7 3731 24859 6.0 5044 30308 5.5 6521 35717 5.0 8264 41325 4.6 9775 45448 85 TABULATED LOCAL HEAT TRANSFER DATA Run Number 45, Q = 0.646 BTU/sft sec X Nu Re Pr Gr Ra 1 7.9 325 8.1 2188 17656 3 7.2 333 7.9 2699 21192 5 6.9 340 7.6 3120 23849 7 6.9 349 7.4 3465 25775 9 6.8 357 7.2 3867 28012 12 6.6 372 6.9 4618 32028 24 6.5 427 5.9 7703 45568 36 6.4 482 5.1 11679 60051 48 6.4 543 4.5 16714 75226 60 6.4 603 4.0 22745 90569 70.5 6.5 654 3.6 28120 101948 TABULATED LOCAL HEAT TRANSFER DATA Run N; imber 46, Q = 0.642 BTU/sft sec X Nu Re Pr Gr Ra 1 9.1 378 6.8 3661 24730 3 7.6 387 6.6 4741 31223 5 7.0 396 6.4 5582 35820 7 6.7 404 6.3 6289 39326 9 6.7 413 6.1 6805 41497 12 7.0 429 5.8 7375 43126 24 6.8 484 5.1 11182 56983 36 6.8 . 543 4.4 15940 71191 48 6.9 604 4.0 21700 85424 60 7.0 664 3.5 27718 98218 70.5 7.1 715 3.3 33303 108565 

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