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Measurement of heat transfer coefficient in concentric and eccentric annuli Choudhury, Wasiuddin 1963

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MEASUREMENT • OF HEAT... TRANSFER COEFFICIENT IN CONCENTRIC AND ECCENTRIC ANNULI by WASIUDDIN CHOUDHURY B.Sc.Eng. , University of Dacca, 1958 A THESIS SUBMITTED IN.PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA APRIL, 1963 i i In presenting this thesis i n p a r t i a l fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shal l make i t freely available for reference and study. .1 further agree that permission for ex-tensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publications of this thesis for f inancial gain shal l not be allowed without my written permission. Department of Mechanical Engineering, The University of Br i t i sh Columbia, Vancouver 8, Canada. A p r i l , 1963. i i i ABSTRACT Heat transfer coefficients were measured at the inner wall of smooth concentric and eccentric annuli. The annuli were formed by an,outer plast ic tube of 3 i n . inside diameter and an inner core cylinder of 1 i n . outside diameter. A i r at room temperature was; allowed to flow through the annuli at a Reynolds number range from 15,000 to 65,000. The measurement of heat transfer coefficient was made by a transient heat transfer test technique. The method consisted i n establishing an i n i t i a l temperature gradient between the f l u i d and a so l id body mounted on the core cylinder by heating, then observing and recording the temperature-time history of the body as i t returned to equilibrium condition with the f l u i d stream. The heat transfer coefficient was calculated from this record. The f i n a l results were presented i n graphical form.showing variations of Nusselt number with Reynolds number. The results of the concentric annulus tests agreed favourably with those predicted by Wiegand (2) and Monrad and Pelton (3)' The effect of eccentricity was to reduce the heat transfer coefficient although the general trend was identical to that in the concentric annulus case. It was observed that the decrease of heat transfer coef-f ic ient was not l inear ly related to the eccentricity of the core cylinder. The effect of eccentricity was more pronounced in the range 0 < e < 0.5 where the value of heat transfer coefficient de-creased considerably. X •ACKNOWLEDGEMENT The author would l ike to thank the many persons whose direct or indirect assistance has made this report possible. He should l ike to mention some of them by name: Professor W. 0. Richmond deserves many thanks for his continued encouragement and assistance during a l l phases of the work. Professors V. J . Modi, N. Epstein) C . - A . Brockely and Mr. J . D. Denton for many valuable discussions during different phases of the project. Professor W . A . Wolfe who suggested the present.topic and through whose i n i t i a l efforts the project was started. F ina l ly the author would l ike to express his thanks for the permission to use the Computing Centre at the University of Br i t i sh Columbia and the National Research Council of Canada for providing funds to make this research possible. i v TABLE OF CONTENTS Page CHAPTER I Introduction.. . . . . . . . . . . . . . . . 1 Definitions of Terms Used 2 Review of 'Literature on-Annuli ' 3 CHAPTER II The Transient Test Technique 9 Discussion of Assumptions . . . . . . 12 CHAPTER III Apparatus 15 instrumentation . . . . . . . . . . . . 2h CHAPTER IV Experimental Procedure 27 Presentation of Results . . . . . . . . . . . . . . . . . . . . 28 Discussion of Results 3^-CHAPTER V Conclusions . . . . . . . . . . . . . . . Mi-Recommendations k-5 BIBLIOGRAPHY k6 APPENDIX I A i r Metering System . . . . . . . . ^9 V Page APPENDIX II Physical Properties and Constants of the Thermal . Capacitors 52 APPENDIX III . Physical Properties of Air 53 APPENDIX TV Sample Calculations 56 APPENDIX V Tables of Experimental Results 57 v i LIST OF FIGURES Page 1. . Schematic Layout.of the Concentric Annulus Tests . . . 19 2. Test Models and Mounting Structures . . . . . . . . . 20 3. Schematic Layout of the Eccentric Annulus Tests . . . 21 k. Method of Supporting Inner Tube . . . . . . . . . . . . . . . 22 5. Photographs of Test Section and Capacitors . 23 6. Thermocouple Circu i t . . . . . . . . . . . . . . . . . . . . 26 7« A Typical Temperature-Time Relation as Obtained on the Recorder . . . . . . . . . . . . . . . . . . . . . . . . . . 29 .8. Heat Transfer at Inner Wall of Annulus, e = 0. . . . . : 30 9« Heat Transfer at Inner Wall of Annulus,.e.= Q'.25 .... .31 10. Heat Transfer at Inner Wall of Annulus, e = 0.50 , . . 32 11. Heat Transfer at Inner Wall of Annulus, e = 1.0 . . . .33 12. Variation of Nusselt Number with Reynolds Number and Eccentricity Parameter . 35 13. Variat ion of Heat Transfer Coefficient with Inner Tube Eccentricity . . . . . 36 l U . Comparison of Heat Transfer Data Obtained by Capacitors No. 1 and No. 2 . . ..................... 38 15« A Typical Semi-Log Plot of Dimensionless Temperature with Time kl 16. Thermocouple Locations in the Capacitor k2 17. Dynamic Viscosity of A i r at Atmospheric Pressure . . . 5^  18. Thermal Conductivity of Air at Atmospheric Pressure. . 55 v i i LIST OF TABLES Page ; I i Details of Orifice 51 II. Properties of Thermal Capacitors 52 III. Biot Number of the Capacitors . . . . . . . . . . . . . . . . . 52 TV. Temperature-Time Data for Capacitor No. 1, e = 0 . . 58 V. Data for Computing Reynolds and Nusselt Numbers for Capacitor No. 1, e = 0 ... 59 VI. Temperature-Time Data for Capacitor No. 1,. e = O.25. 60 VII. Data for Computing Reynolds and Nusselt Numbers for Capacitor No. 1/ e .= 0.25 . . . . . . . . . . . . . 6 l VIII. Temperature-Time Data for Capacitor No..1, e = 0.50. 62 IX. Data for Computing Reynolds and Nusselt Numbers for Capacitor No. 1, e = 0.50 63 X. Temperature-Time Data for Capacitor No. 1, e = 1.0 .. 6k XI. Data for Computing Reynolds and Nusselt Numbers for Capacitor No. 1, e = 1.0 65 XII. Temperature-Time Data for Capacitor No. 2, e = 0 . . 66 XIII. Data for Computing Reynolds and Nusselt Numbers for Capacitor No..2,. e •= 0 . . . . . . . . . . . . . . . . . 67 v i i i LIST OF SYMBOLS a, a"*" - constants p A - surface area of the capacitor, f t C - heat capacity of the capacitor, Btu/F 0^ - specific heat of a i r , Btu/ lb . F C^ "*" '- specific heat of the capacitor, Btu/lb F - heat capacity of the supporting p las t ic , Btu/F D-^  - outer diameter of the inner cylinder, ft., LV> - inner diameter of the outer tube, f t k x flow area , D e - hydraulic equivalent .diameter, w e t t e d perimeter f t k x flow area , D n -Nusse l t equivalent diameter, h e a t e d perimeter f t e - eccentricity parameter G - mass velocity, lb /hr f t 2 h - average heat transfer coefficient, Btu/hr ft' F 1^ - maximum velocity head, i n . of water h s - stat ic pressure head, i n . of water h w - pressure drop across or i f i ce , i n . of water K - thermal conductivity of f l u i d , Btu/hr ft F - thermal conductivity of plast ic support, Btu/hr f t F Kg - thermal conductivity of capacitor material, Btu/hr f t -F 1 m, m - constants constants hDe ~K - Nusselt number, I X N - Prandtl number, • • 1 Pr GDe W^e - Reynolds number, r-j_ - outer radius of the inner cylinder, f t r^ - inner radius of the outer tube, f t r - radius at the point of maximum velocity, f t m T - temperature of the capacitor at any instant, chart div. T . - temperature of a i r , chart div. or F a T Q - temperature of capacitor at zero time,.chart div . T* - dimensionless temperature V a V g - average flow velocity, . f t / h r V m a x - maximum point velocity, f t /hr w - mass of the capacitor, lb 9 • - time,. sec A - slope, l / h r - dynamic viscosity of a i r , lb /hr f t ^M.w - dynamic viscosity of a ir at the so l id wall , lb /hr f t ^ - density of a i r , l b / f t 'XQ - time constant of the capacitor,.sec . - time constant of the instrument, sec - functions 1 CHAPTER I INTRODUCTION The purpose of this investigation was to measure the heat transfer coefficients in.an annulus formed by placing a smooth cylinder at various positions inside a c ircular tube with f l u i d flowing tur-bulently along the longitudinal axis of the cylinder. The heat flow was through the inner cylinder surface and the variations of heat transfer coefficients due to different positions of the inner cylinder were studied. The investigation was l imited to the Reynolds number range from 15,000 to 65,000. This problem is of interest because of developments in the peaceful uses of atomic energy which necessitates a knowledge of heat transfer coefficients i n certain types of odd-shaped flow passages. In the present designs of heat exchangers and nuclear reactors, the coolant flows in longitudinal passages formed by the spaces between para l l e l rods or tubes in* closely spaced arrays within a cy l indr ica l container. In gas cooled nuclear reactors, arrays of fuel bearing rods are cooled by gas flowing para l l e l to the longitudinal axes of the rods. This general configuration provides increased heat transfer surface to compensate for the reduced heat transfer coefficients as-sociated with gaseous flows. This type of flow geometry has received very l i t t l e attention from research workers. This has resulted i n the use of heat transfer coefficients in design which are not always safe and accurate. An experimental programme was undertaken for the very simplest case of this geometry, namely an annular cross-section. 2 The usual method of determining the heat transfer coefficients requires the measurement of the temperature of the heat transfer me-dium, and of the heat, transfer surface under steady-state conditions. It is. d i f f i c u l t to measure the,surface temperature accurately so that steady-state methods require elaborate apparatus and instrumentation. To obviate these d i f f i cu l t i e s a non-stationary method has been used i n this investigation. This method, the transient heat transfer test technique ( l )* is based on the cooling.of a body having fixed thermo-physical properties under a step-change of temperature. The under-lying theory of this technique w i l l be discussed i n . d e t a i l in the next chapter. Using this technique results were.obtained .for heat transfer in the concentric annulus. These results were compared with correla-tions given by Wiegand (2) and Monrad and Pelton (3). Results for the eccentric annulus were also obtained in a similar manner. These lat ter results are of interest because of the paucity.of data for the eccen-t r i c annulus. DEFINITIONS OF TERMS USED Annulus An annulus is defined as the flow passage formed by placing a cy l indr ica l core inside a c ircular tube with f l u i d flowing along the longitudinal axis of the core. When the cy l indr ica l core is at the centre of the outer tube the resulting flow cross-section is termed a concentric annulus. • An eccentric annulus is formed by placing the core cylinder at positions other than the centre of the outer, tube. * Numbers in parentheses refer to bibliography at the end of the Thesis. 3 Thermal Capacitor The heated so l id body used in.the exper-iment is cal led the thermal capacitor cylinder. For the sake.of bre-v i ty in the subsequent chapters this w i l l be cal led the capacitor. REVIEW OF LITERATURE ON ANNULI Concentric Annulus A large number of experiments has been carried out on heat transfer in turbulent flow in a concentric annulus. Most investigators attacked the problem by dimensional analysis and presented equation basical ly of the form, m n D %u - a (%e) ttpr) 01 C^) — - [ l ] hD e where, N|gu = Nusselt number, • -— GD e . 1 L 0 = Reynolds number, —— ite ^/\A Np^ , = Prandtl number, ^~ a, m.and n are constants which were determined experimentally. This equation is of the same form as that for conduit flow, 1 1 / D 2 %u = a (%e)m ( N P r ) n > w i t h t h e addition of the term, 01  ^ to include the dimensional characteristic of the annulus. Jordan (2U), i n 1909> studied the heat transfer from the inner wall of two ver t i ca l annuli of different dimensions. The annuli were formed by placing a copper pipe inside a cast iron casing. The diameter ratios were 1.25 and 1.^7• A ir at temperatures varying from 2^0 F to 700 F was passed through the inner tube and water at temperatures up ."to 200 F was circulated through the annular space. The test section was hO f t . long and i t was placed direct ly after an elbow. The experimental results covered a Reynolds number range from-7,000 to 95 ,000 . Thompson, and Foust ( 25 ) , in 19^+0, investigated the heat transfer characteristics in two double-pipe heat exchangers. One of them was made of one inch pyrex tubing jacketed with two inch iron pipe; the other was two inch pyrex tubing i n a three inch pipe. Cold water at 50 F to 95 F flowed through the annular space and :sfceam.!. was con-densed at the inner pyrex tube. The test section was 10 f t . long and was placed direct ly after an elbow. The heat transfer, coefficient was determined from the overall resistance by assuming that the coefficient was a function of water velocity in the annulus only. The experimental data was taken at Reynolds number greater than 100,000. Foust and Christ ian ( 4 ) , i n the same year, investigated the heat transfer coefficients i n annuli having various thicknesses of the annular passage. The annulus was formed by assembling thin walled;, copper tubing inside standard iron pipe. The diameter rat io ranged from 1.20 to 2 . 5 6 . Water flowed through the annular space and steam was condensed inside the copper tube. The test section was 8 .66 f t . long with no provision for a calming section. The heat transfer coef-f ic ient was calculated by graphical dif ferentiat ion. The investigation covered a Reynolds number range from 3>000 to 60 ,000 . Thet equation recommended was, ? . . . o < ? ) " fc)" (1) H 5 This equation was based on,Nusselt's equivalent diameter, D n defined as, D _ h x Flow area  n Heated Perimeter The authors also found.that the results could be correlated by using the conventional equivalent diameter, based on the wetted perimeter, by the following equation, * - « ( J ) " ( j) H Zebran (26) i n the same year also studied the heat transfer characteristics at the inner wall of an annulus. The•core cylinder was e l ec tr i ca l ly heated and a ir was forced through the annular passage. Five different diameter ratios were used ranging from-.1.18 to 2.72. Provisions were made for a calming-section which varied between 8 and 125 equivalent diameters. The heat transfer coefficients were ca l -culated by measuring the core wall temperature and the temperature of the a ir stream. The experimental data covered the Reynolds number range from 2,600 to 120,000. . Monrad and.Pelton (3) i n the same year presented experimental results of heat transfer coefficients i n three different annuli with two different f luids flowing i n turbulent flow. The diameter ratios were I.65, 2.^5 and 17 for heat transfer from.inner wall , and I.85 for heat transfer from outer wal l . The test section.was 6.5 f t . long with a calming section at the entrance. The investigation covered a Reynolds number range from 12,000 to 220,000. The experimental data 6 for heat transfer at the inner wall were correlated.by the equation, • / \ ° - 8 / \ n / \ 0.53 where n = O.h for heating and.0.3' for cooling. The properties were evaluated at the main stream temperature. It was also suggested, on the assumption that the relative skin f r i c t i o n on the two s'urfaces was the same for turbulent as for laminar flow, that the constant term, AM 0 - 5 3 0.02 I — I , i n equation |_UJ could be modified for better ac-+ n o « l 21n D 2 / D l - ( D2 / D l ) 2 + 1 curacy to 0.023 [ ' *—' : D2/Dx - D X/D 2 - 2(D 2 /D X) I n D 2 / Y ) ± It is to be noted here that the agreement between the equa-tions of Foust and Christ ian, and Monrad and Pelton is very poor. As Barrow (5) pointed out, for diameter ratios 16 and 2.k respectively, the former investigators predict results 6.0 times and 2.2 times greater than those predicted by the lat ter investigators. Mueller (6), also in the same year, investigated heat trans-fer from a fine wire placed inside a tube. This was a very special case of annuli because with such a small core and large thickness of . annulus a combination of cross flow and para l l e l flow is l i k e l y to exist . Therefore, his correlations could not be compared with an annulus of the proportion used in this investigation. Davis (7), i n 19^ 3, presented another equation for heat trans-fer from inner.wall with turbulent flow of f lu ids . The equation was obtained by dimensional analysis and was valid for diameter ratios from 1.18 to 6,800. The suggested equation was, . / \ 0.8 / x 1/3 , v 0.1k' , x 0.15 The constants i n this equation were determined by comparison with the experimental data of other investigators. McMillen and Larson (27), i n 19kk, obtained annular heat transfer coefficients i n a double pipe heat exchanger. The test section consisted of four passes having different annular spaces with brass or steel tube placed inside a standard iron pipe. The diameter ratios were 1.245', 1-3, 1.532 and 1-970. The length of the test.sec-tion.was 11 f t . There were no calming sections at the entrance of each pass. The coefficients were calculated by assuming that a l l the thermal resistances except that of the film were constant. The ex-perimental data covered a Reynolds number, range from 10,000 to 100,000. The best correlation of data was given by the equation, = 0.0305 —- [6] Wiegand (2), i n 19^ -5, made a detailed study of a l l the ex-perimental data on heat transfer from the inner wall. The best cor-relation f i t t i n g these data was found to be, The equation was va l i d for Reynolds number greater than 10,000 where the properties were calculated at the bulk temperature. 8 Apart from dimensional analysis, this problem has also been attempted by establishing an analogy between f l u i d f r i c t i o n and heat transfer. In 1951> Mizushina (10) investigated this approach but the analysis was quite d i f f i c u l t due to lack of similarity between the velocity and temperature profiles when heat was flowing from .the inner wall only. Using' many assumptions Mizushina derived an equation for the heat transfer coefficient which was very complicated for general use. Barrow ( l l ) , i n 1961, presented a semi-theoretical solution by considering the transfer of heat and transfer of momentum. The sol-ution was applicable for fluids having a Prandtl number equal to unity. Eccentric Annulus Heat transfer i n eccentric annuli has received very l i t t l e attention. Deissler and Taylor ( l 2 i ) , i n 1955> made a theoretical analysis of f u l l y developed turbulent heat transfer i n an eccentric annulus. It was found that the average Wusselt number decreased as the eccentricity of the inner tube was increased. It was also shown that the Wusselt numbers for a concentric annulus were sli g h t l y higher than those for a tube with an equivalent diameter. In 1962, Leung, Kays and Reynolds (13) presented heat transfer data i n concentric and eccentric annuli based on a theoret-i c a l study as well as an experimental investigation. For the eccentric annulus the diameter ratios ranged from 2 .0 to 3»92 and for the con-centric annulus from 2 .0 to 5»2. Air was flowing through the annulus under f u l l y developed turbulent flow conditions. The experimental results covered a Reynolds number range from 10,000 to 150 ,000. 9 CHAPTER II THE TRANSIENT TEST TECHNIQUE The transient heat transfer test technique as used in this investigation has also .been used by other investigators in different types of flow situations. London, Boelter and Nottage ( l ) proposed this method i n 19^1, Eber (l^) used this method to obtain.data on heat transfer characteristics i n superonsic flow over cones, and Fischer and Norris (15) determined the convective heat transfer coefficient by an analysis of skin temperature measurements on the nose of a V-2 rocket i n f l i g h t . In the past decade Garbett (16) used the transient technique to determine the heat transfer coefficient from bodies i n high velocity flow. Kays, London and Lo (l?) demonstrated success-f u l l y i t s usefullness in determining the heat transfer coefficient for gas flows normal to tube banks. •Basical ly the transient technique for determining heat transfer coefficient consists i n establishing an i n i t i a l temperature potential between the f l u i d and the body by heating.or cooling, then observing and recording the temperature-time history of the body as i t returns towards equilibrium with the f l u i d stream. The heat transfer coefficient is calculated from this record. The simplest case of transient heating or cooling of a body i s one i n which the internal resistance of the body is negligibly small i n comparison with the thermal resistance of the sol id-f luid , interface. This is because such a system permits the lumping of a l l the resistances to heat transfer at the boundary of the so l id body. 10 This ideal i sat ion. i s based on the assumption that the material of the capacitor has i n f i n i t e l y large thermal conductivity. .Many transient heat flow problems can be solved with' reasonable accuracy by assuming that the internal conductive resistance is so small that the temper-ature throughout the body is uniform at any instant of time. A quantitative measure of the relat ive importance of the two resistances can be expressed by a dimensionless modulus, cal led the Biot number> which is defined as, = B L KS where, h = surface coefficient of heat transfer =- , , . . . , j . . Volume of the body L = a characteristic length, — — Surface area K s = thermal conductivity of the body As Kreii^h (l8) pointed out, for bodies whose shape resembles a plate, a cylinder, or a sphere, the error introduced by assuming that the internal temperature at any instant is uniform w i l l be less than 5 per cent when the internal resistance is less than 10 per cent of the external surface resistance. In other words this w i l l hold when N B i < 0 . 1 . Keeping in mind the above requirement, the capacitor of the present experiment was made of copper which has one of the highest thermal conductivities amongst the naturally available metals. For a capacitor of this metal, the rate of temperature change w i l l depend 11 only on the average surface heat transfer-coefficient which varies mainly with the f lu id properties, the flow conditions, and the flow geometry. The theoretical equation describing the energy balance on the cooling capacitor under this idealised condition can be formulated as, The change of internal energy of the thermal capacitor dur-ing a small interval of time -C dT = hA (T - Ta) d'9 The net heat flow from the thermal capacitor to the f l u i d stream during the same interval of time where, C = heat capacity of the capacitor A = surface area in contact with f l u i d stream 9 = time T = temperature of the cooling capacitor at any 9 Ta = temperature of the f l u i d stream The minus sign on the l e f t hand side in equation [_Q~]' indicates that the internal energy decreased when T > Ta as was the case in this experiment. In solving equation [jf) , C, A and Ta are assumed constant. For small variation of. T and constant flow, h w i l l be essentially constant. The solution becomes, T - Ta To - Ta exp hA 9 03 12 where, To = temperature of the capacitor•at.9 = 0. This equation can be rewritten in the following form, log„ T* = I -r ~ ] 6 [10] where, T* = 3e . \ • C T - Ta To - Ta If the assumption i s legitimate then a plot of l o g g T* as the ordinate and. 9 as the abscissa would y ie ld a straight l ine the slope of which w i l l be — From this slope the value of h can be calculated. DISCUSSION OF ASSUMPTIONS In establishing and solving equation QjJ several assumptions have been made. - A . c r i t i c a l evaluation of these assumptions is nec-essary at this stage to support the appl icabi l i ty of the transient test technique to heat transfer in annular flow. 1. The assumption involving the ideal isat ion that a l l re-sistances to heat transfer are lumped at the so l id f l u i d interface is the key point to the whole analysis. Accuracy requirements for a given problem.determine whether or not this approach.is useful. In most heat transfer experiments, results with an error of plus or minus 25 per cent may be considered very good. However, Kays,, London and Lo (17) claimed that their experimental error was plus or minus 5 P e r cent where they used a similar method for the cross-flow heat exchanger. It is. expected that in the present experiment the error would be con-fined to the same l i m i t . 13 2. Heat transfer is assumed to be by forced .convection only. In a f u l l y developed turbulent flow free convection effects are almost negligible. However, in viscous flow with a large'annular gap the poss ib i l i ty of free convection effects cannot be to ta l ly ignored. . In this experiment the flow conditions were always turbulent and therefore free convection effects can be neglected. It is also assumed that radiation effects are negligible. For small temperature difference at low temperature leve l with pol -ished model surfaces this effect would be very small. But for high temperatures, radiation loss can be significant which means that the heat transfer coefficient as determined by this method would contain a contribution f rom the radiation coeff icient. Temperatures of about 60 to 70 F above ambient can be considered low enough so as not to cause any serious error due to radiation. The present experiment was conducted in this range of temperature. 3• The temperature difference i n the radia l direction inside the capacitor and i n the axial direction along i t s length is negl i -gible . This can be achieved by having a thermal capacitor of very high thermal conductivity and re lat ive ly thin wall thickness. k. Heat loss to the supporting structure of the model is assumed very small. -As Garbett (16) noted in his investigation, the mounting structure sometimes provide additional sources of heat loss. For example, during a cooling process the capacitor cools much faster than the supporting medium so that towards the end of the transient cooling heat may flow from the supports to the capacitor. This effect Ik can be best-reduced .by keeping the area of contact between capacitor and.support small and using supporting material having a small (K^C^) product, where, . = thermal conductivity of the supporting material = heat capacity of the supporting material 5• The average value of the specific heat of the material of the capacitor is constant. The specific heat of•copper is essen-t i a l l y constant between temperatures-'70 to 150F. The mass of the capacitor is also constant;, and therefore the heat capacity is constant. 6. The bulk temperature of the f l u i d stream is constant during the test. To verify this a i r temperature was measured con-tinuously at the outlet of the test .section for the duration of one ..complete run. It was found that the variation was not more than 1 F which was less than 1.5 per cent of the ambient temperature. 15 CHAPTER III APPARATUS The apparatus was designed to measure the following: (a) the a ir flow rate (b) the temperature and pressure of a ir (c) the temperature-time history of the capacitor Two different arrangements of apparatus were used. One for the study of the concentric annulus and another for the eccentric annulus. In Figure 1, a schematic layout of the apparatus used for the concentric annulus is. shown. The arrangement consisted of a test section, a mixing box and a centrifugal blower. The test section, shown in Figure 2, was made of transparent plast ic outer tube having an inside diameter, Uq = 3 i n . , and a wall thickness of l/k i n . The inner cylinder on which the capacitor wasi mounted had an outside dia-meter, = 1 i n . Air at room temperature was blown through the test section by a l/3 hp centrifugal type blower with a speed rating of 3^50 rpm. The a i r flow rate was determined by measuring the maximum velocity in the annulus by a pitot static tube placed at the downstream end of the test section and connected to a micro-manometer. A mixing box between the blower and the test section acted as a surge tank and tended to smooth pulsations in the a ir stream. 16 The temperature variation of the cooling capacitor was measured by a copper-constantan thermocouple connected to an .automatic recorder to obtain a temperature-time history. The apparatus'" used for the eccentric annulus test is shown schematically in Figure 3« The same test section and mixing box were used i n this arrangement but a higher capacity blower was used to allow an or i f i ce meter to be used for measuring a ir flow. In the ec-centric annulus the a ir flow rate could not be determined by pitot tube as in the case of concentric annulus because of the non-symetrical. fom of velocity prof i le in an eccentric annulus. The flow rate was therefore measured by a f la t plate, square edged or i f ice placed up-stream of the test section. The large pressure drop across the or i f ice required the higher capacity blower to cover the same range of Reynolds number as was obtained in the concentric annulus flow. A detailed description of the two a ir metering systems i s given i n Appendix I . The blower used in the second arrangement was a 3/h hp centrifugal type blower with a speed rating of 3^50 rpm. A i r temp-erature was measured by a mercury in.glass thermometer graduated to 1 F and also by a copper-constantan thermocouple placed in the mixing box. The change i n the flow Reynolds number was accomplished by thrott l ing the a ir flow at the in le t side of the fan. The length of the test section was selected to give a f u l l y developed hydrodynamic flow. There is s t i l l a considerable doubt as to the exact length after which the flow is f u l l y established in an annulus. One group of investigators, M i l l e r , Byrnes and Benforado (19) 17 claimed that i n a concentric annulus the velocity prof i le is establish-ed i n twenty equivalent diameters. In the present investigation, this figure was taken as the design cr i ter ion to select the length of the test section. It was also found that there was no re l iable information available as to the entry length required for the establishment of a f u l l y developed flow in an eccentric annulus. However, i t was assumed that the entry length.in both concentric and eccentric annuli was of the same order of magnitude. The heat transfer characteristics of the capacitor was studied by recording i t s temperature-time history during a transient cooling period. Two capacitors were used. The f i r s t one, shown in Figure 2, was 2.125 i n . long with an outside diameter of 1 i n . and an inside diameter of.0.776 i n . The capacitor was made from so l id copper rod d r i l l e d to the required inside diameter and faced to the desired length. A small hole was d r i l l e d in the wall of the capacitor to provide space for the thermocouple junction. In selecting the size of the capacitor, care was taken to comply with t h e B i o t number re-quirements and other assumptions made i n Chapter I I . A second capacitor was made of the same material as the f i r s t but was smaller i n length and mass. It was 1.1+703 i n . long with an outside diameter of 1 i n . and an inside diameter of 0-7175 i n . Figure 2 shows the dimensions of this capacitor. A few tests were made with this capacitor to show that the length of the capacitors had l i t t l e effect on the results obtained. Further details of the comparison are given in Chapter IV. 18 The capacitor was mounted in the test section at the down-stream end of the inner cylinder. The mounting structure consisted of two plast ic end pieces, one on each side of the capacitor. The plast ic sections had the same outside diameter as the capacitor so that f i n a l assembled form has a smooth outside surface. For the concentric annulus the remaining length of the, inner cylinder, upstream of the capacitor, consisted of a so l id plast ic rod having an outside diameter of 1 i n . Any small deflection due to the weight of the rod could be ignored considering the large thickness of the annular passage. But in the eccentric annulus, especially when the inner cylinder was closer to the wall of the outer tube, any small deflection would cause distortion i n the flow pattern. To minimise this effect the inner cylinder was madeiof thin walled aluminium tubing: .with plast ic end pieces to allow for the mounting of the capacitor. The overal l de-f lect ion of the-aluminum tube was less than 0.010 i n . Supports at the two ends were used to position the inner cylinder within the tube. To al ign the tube accurately, templates were • made from aluminium' plate. These templates had an outside diameter of 3 i n . • with an one inch hole made at the required eccentric i t ies , namely, e = 0, e = 0.25 and e = 0.50. Having aligned the cylinder with these templates, two thin metallic spiders crossing at right angles were placed at the two ends of the cylinder. Right angled grooves were made, at both ends of the inner cylinder to locate the spiders; The inner cylinder with the spiders was then easi ly s l i d into the outer tube. The same procedure was followed for the eccentric annulus but in this case the l ine of intersection of the two spiders RECORDER MICRO-MANOMETER PIG. 1. SCHEMATIC LAYOUT OP THE CONCENTRIC ANNULUS TESTS' H ^PLASTIC END PIECESN 4 r ?0 r PLASTIC FOR CAPACT1 rso.-ij e a 0 CAPACITOR 1.5 - — V ALUMINIUM TUBE FOR ALL OJT-HER RUNS 2.0 2.1250 FOR #1 * it v 'l.4703 FOR #2 ' SUPPORTING ROD FOR CAPACITORS 1.0 2.1250 d. CAPACITOR NO. 1 HERMOCOUPLE HOLE 0.7 1.4703 ! THERMOCOUPLE HOLE CAPACITOR NO. 2 NOTE! ALL DIMENSIONS IN INCHES FIG. 2. TEST MODELS AND MOUNTING STRUCTURES ro o RECORDER ORIFICE7 THERMOMETER THERMOCOUPLES /FLOW SCREEN THERMAL CAPACITOR-FLOW TEST SECTION 44.50 i n . MANOMETERS FIG. 3. SCHEMATIC LAYOUT OF THE ECCENTRIC ANNULUS TESTS TEST SECTION SHOWING SUPPORT AT ONE END. OTHER END SIMILAR SECTION A-A 4. METHOD OP SUPPORTING INNER TUBE FOR e = 0 (a) PHOTOGRAPH OP TEST SECTION (b) PHOTOGRAPH OP CAPACITORS PHOTOGRAPHS OP TEST SECTION AND CAPACITORS 2k was shifted to give the desired eccentricity of the inner cylinder. The supporting device for the concentric annulus is shown i n Figure k. Photographs of the test section and the capacitors are shown i n Figure 5• INSTRUMENTATION ' As mentioned i n the earlier chapter, the heat transfer co-efficient was determined from a record of the temperature-time history of the cooling capacitor. Since the temperature of the capacitor was decreasing with time, a measuring probe capable of fast response to a changing temperature was desired. An ideal temperature measuring probe would f a i t h f u l l y respond to any change regardless of the time rate of temperature change of the capacitor. The thermocouple was the obvious choice for this type of measurement because of the f o l -lowing reasons: .' The time constant of the thermocouple wire i s a function of the ratio of heat.capacity to i t s surface area. The thermocouple junctions have small mass and hence small thermal capacity. The area of contact is comparatively large due to soldering of the probe to the capacitor. Therefore, the thermocouple wire has a small time constant and follows closely any temperature variation i n the capacitor. In the range of the experiment, which was from 70 to 1^0 F, the electro-motive force of the thermocouple is practically a linear function of the temperature. Thus i t was possible to plot the dimen-sionless temperature T* in equation N-Ol i n terms of the emf difference 25 instead of the temperature difference. This simplified not only the computation but also the cal ibration of the recording instrument. The copper-constantan thermocouple used for measuring the capacitor temperature and the a ir temperature in the mixing box was 26 gauge B & S, so l id wire with polyvinyl insulation. The temperature-time variation of the capacitor was recorded on, the chart of a B r i s t o l recorder. The range of the recorder was adjusted to 0 to 2.5 mi l l ivo l t s and the chart speed was kept constant at 3/8 in . .per min. It was believed that the Br i s to l recorder had a faster response than that of the capacitor so any error due to instru-ment lag was negi l ig ib le . A quantitative estimate of the relat ive error w i l l "be presented in the discussion of results , Chapter XV. This error, according to Garbett (l6), i s the rat io of the instrument time constant to the capacitor time constant. Two thermocouples were used i n the experimental arrangement with a selector switch to connect one thermocouple to the recorder at any one time. A schematic diagram of the thermocouple c i rcu i t with related instrumentation is shown in Figure 6. 26 r V REP. JUNCTION 32 P .JUNCTION BOX COPPER : CONSTANTAN T -T.C. TO CAPACITOR T a -T.C. TO MIXING BOX .SELECTOR SWITCH BRISTOL RECORDER PIG. 6. THERMOCOUPLE CIRCUIT 27 CHAPTER IV EXPERIMENTAL PROCEDURE For each set of experiments the inner tube was f i r s t aligned to the particular eccentricity for which the test was to be made. The capacitor was then heated to 60 to 70 F above the ambient temperature by an external heat source consisting of an e lectr ic soldering iron with an aluminium extension piece to reach the capacitor inside the tube. The heat source was then withdrawn, the recorder and the a ir fan motor switch were turned on. As the capacitor was cooling towards equilibrium in the f l u i d stream, a continuous record of temperature against time was made in the recorder. This record was taken for . approximately 130 seconds. The selector switch was then turned on to the a ir temperature thermocouple and a record of this temperature was also made on the same chart. About the middle of each run; or i f i ce pressure drop, static pressure and temperature of a ir at the in le t side of the or i f ice and the a ir velocity were recorded. When one test was complete, the flow.rate was changed by thrott l ing the a ir at the in let to the fan. The same procedure was then repeated t i l l a Reynolds number range from 15,000 to 65,000 was covered. A tota l of about 13 to 16 runs were taken for each position of the annulus. Four positions of the annulus were investigated. Each posi-t ion was designated by the eccentricity parameter, e, defined by the following•relation: e _ Distance between the centres of the two cylinders r2 " r l 28 The four positions were thus, e = 0, e = 0.25, e = 0.50, and e = 1.0. PRESENTATION•OF RESULTS The temperature time history of the capacitor as obtained for a typical run is shown.in Figure 7. For the purposes of comp-utation a l l the.experimental data were entered in tables; given in Appendix V. The f i n a l results have been presented in graphical form with dimensionless parameters on log-log co-orindates. This presenta-tion was of the following form: NNu = i 2 S 2 ( W Re) In the present experiment, the a ir was at room temperature which at times varied between 70 F and 90 F . The Prandtl number for a ir i n this temperature range is pract ica l ly constant. Therefore, a log-log plot of Nusselt numbers as a function of Reynolds numbers only have' been presented. In Figure 8, the variation of Nusselt number with Reynolds number has been shown for an annulus with eccentricity parameter equal to zero. In the same sheet equations of other investigators such as Wiegand (2), Monrad and Pelton (3) and Foust and Christ ian (k) have also been plotted. The results of the eccentric annulus for three different eccentricity parameters, e = 0.25, e = 0.50 and e = 1.0 are shown i n Figures 9> 10 and 11 respectively. In each case, the best l ine correlating the data.is also shown. 29 -o =e— p 1 ,—1 _ " 0 0 0 hi n n n r.t o O o ri n n f* a o o rti o o O ; ( , r> o -T m o S 5 n n 7? C a w No. R5000 5 QlMI_J> CHUT No. RSOOO 8 l_lu3Z^  CHUT No. RSOOO O [ZSB PIG. 7. A TYPICAL TEMPERATURE-TIME RELATION AS OBTAINED ON THE RECORDER 30 &5 tt w pp E-i H i W CO ra &5 300 270 240 210 180 150 120 90 60 30, WIEGA iND CHRISTIAN ) / ^ ^ ^ ^ ^MONRAD AND I 'ELTON REYNOLDS NUMBER, N R E FIG.. 8. HEAT TRANSFER AT INNER WALL OF ANNULUS, e = 0 31 300 270 240 210 60 1.3xl0 4 2 3 4 5 6 7xl0 4 REYNOLDS NUMBER, N R e pIG. 9. HEAT TRANSFER AT INNER WALL OF ANNULUS, e = 0.25 32 300 270 240 210 180 60 1.3xl0 4 y 2 3 4 5 6 7xl0 4 REYNOLDS NUMBER, N R PIG. 10. HEAT TRANSFER AT INNER WALL OP ANNULUS, e = 0.50 33 300 • 270 • 240 • 210 • 180 -150 1.3x10* 2 3 4 5 6 7x10 REYNOLDS NUMBER, N R e PIG. 11. HEAT TRANSFER AT INNER WALL OP ANNULUS, e =1.0 3^ In Figure 12, the variat ion of Nusselt number with Reynolds number and eccentricity parameter is shown. It is interesting to note that the average Nusselt number decreased with increasing annulus eccentricity. In•Figure 13, this decreasing trend of Nusselt number with eccentricity parameter has been plotted for a fixed Reynolds number. DISCUSSION OF RESULTS This section includes a discussion of a l l the experimental results and an evaluation of the probable errors involved in different parts of the investigation. Experimental data for the average heat transfer coefficient in a concentric annulus are shown in Figure 8. Since a l l these data were obtained in almost isothermal condition, the Prandtl number of the a ir was always constant. The equations of Wiegand, Qi-J of Monrad and Pelton and of Foust and Christ ian are plotted in the same sheet to compare the experimental results . It is noted that the agreement is very good indeed with equations (V) and QJ-J . The equation of Foust and Christ ian predicts much higher values of heat transfer coefficients although the trend shown is the same. .Foust and Christ ian obtained their water side coefficients by means of a semi-graphical method which resulted in steam coefficients which were very high so that their results have been questioned by latter investigators (5, 21). The equation Qjf} of Davis (7) was not considered a proper comparison with the present data because i t is based on the inside 35 270 240 o e = 0.0 • e = 0.25 A e = 0.50 O e = 1.0 1.3x10* 2 3 4 5 6 7x104 REYNOLDS NUMBER, N R -PIG. 12. VARIATION OP NUSSELT NUMBER WITH REYNOLDS NUMBER AND ECCENTRICITY PARAMETER 0.0 0.25 0.5 0.75 1.0 ECCENTRICITY PARAMETER, e PIG. 13. VARIATION OP HEAT TRANSFER COEFFICIENT WITH INNER TUBE ECCENTRICITY 37 diameter of the annulus. Davis agreed that when heat transfer occured only at the inner surface the use of inside diameter in the Nusselt and Reynolds numbers was reasonable. However, as Barrow (5) pointed out the computation of Reynolds number based on the inside wall d ia-meter can in no way be considered as a measure of the kind of flow.. Also the f r i c t i o n in annuli results from resistance at both the inner and outer wall of the annulus. Therefore, the most reasonable defin-i t i o n of Reynolds number is the one based on the conventional equi-valent diameter, D e . The heat transfer coefficients presented here are claimed to be the average value in a f u l l y developed turbulent flow. A question may arise as to the v a l i d i t y of this statement on the grounds that the thermal boundary layer was not completely developed in the length of the capacitor. As mentioned in Chapter II the transient technique does measure the average heat transfer coefficient, provided a l l assumptions have been sat is f ied. To just i fy this statement a second experiment was carried out for the same annulus but with a capacitor of different length. Capacitor No. 2 was thus designed with a length of I.U703 i n . thereby having a reduction in length of 30 per cent of that of Capacitor No. 1. Figure 2 shows the details of the second capacitor. The heat transfer data obtained by using both these capacitors are plotted in Figure lh. The good agreement be-tween these results indicates that the earl ier statement was reasonable. Kays, London and Lo (17) pointed out that the transient technique is not adequate for laminar flow because of the tendency 38 fe fe EH CO •B 300 270 240 210 180 150 120 90 60 30 C o r> ) di ii c > o o • 0 • ! / i O CAP • CAP ACITOR N ACITOR N 0. 1 0. 2 7x10 REYNOLDS NUMBER, N R e PIG. 14. COMPARISON OP HEAT TRANSFER DATA OBTAINED BY CAPACITORS NO. 1 AND NO. 2 39 of the temperature prof i le to extend completely across the flow stream rather than being confined to a re la t ive ly thin boundary layer such as in turbulent flow. This is why the transient technique gives' the average values of heat transfer coefficients provided the flow is f u l l y turbulent. In addition, the results shown in Figure 8 also suggest that the data obtained in . th i s investigation are the average heat'transfer coefficients. Refering to Figure 8 again, i t may be noted that the ex-perimental, point at the lowest Reynolds number tends to be higher than that predicted by the equations of Weigand and Monrad and Pelton. It i s believed that at very low Reynolds number the cooling rate of the' capacitor becomes very slow and hence towards the end of the cooling period the capacitor is l i k e l y to loose heat to the sur-rounding supports at each ends. At higher Reynolds number this loss is negligible because the capacitor cools very quickly. In Figure 12, the variation of heat transfer coefficient with Reynolds number and eccentricity parameter is shown. It was found that the average Nusselt number decreased as the eccentricity was increased. This trend is in. good agreement with Deissler and Taylor's (12) theoretical analysis. The best l ine f i t t i n g the data has also been drawn in each case and i t is noted that the slope of the lines is nearly constant for a l l eccentricit ies of the annulus. It was observed that the tota l decrease in Nusselt number was 36 per cent at the highest Reynolds number and about 1+0 peri.cent at the lowest Reynolds.number. . Figure 13, shows this variation at four ^0 different Reynolds number. The rate of decrease of heat transfer co-eff icient was found to be greater within the f i r s t 50 P e r cent of the eccentricity. Between 50 and 100 per cent eccentricity the var-iat ion of heat transfer coefficient was much.less. It was pointed out i n Chapter II that in the solution of equation (V) the heat transfer coefficient and the heat capapcity of the test model were assumed constant. Considering the fact that the semi-log plot of equation [loj from the experimental data followed the expected straight- l ine relationship, these assumptions seem to be jus t i f i ed . A typical semi-log plot of •equation (jLOj is shown in Figure 15• However, any definite departure from the l inear relation would necessarily make the above assumptions questionable. The con-stancy of the heat capacity of the model w i l l depend on the variation of the specific heat of copper with temperature. The temperature range in the present investigation was between 70 F and ikO F . It is believed that the variation of specific heat i n this temperature range was neg-l i g i b l e . The instrument lag effect may cause an error in the measure-ment of heat transfer coefficient especially when the data recorded were those of a transient state. As mentioned in Chapter III , the relat ive error in the calculated value of the heat transfer coeff i -cient is given by the following relat ion: Relative error = -— x 100 per cent -^c i l l 0.4 0 10 20 30 40 50 60 70 TIME, 9 (sec) PIG. 15. TYPICAL SEMI-LOG PLOT OF DIMENSIONLESS TEMPERATURE WITH TIME k2 w h e r e , = time constant of the recording instrument. T c = time constant of the capacitor model. The time constant of the recording instrument was obtained by apply-ing a step change in the input. This was found to be 2.5 sec. The time constant of the capacitor was given by the inverse of the slope of the semi-log plot shown in Figure 15. The lowest time constant of the capacitor during the entire experiment was that of Capacitor No. 1 during Run 13• This was found to be 9^-5 sec. Therefore, the maximum relative error was 2.67 per cent. In Chapter II itwas assumed that the temperature gradient ,inside the capacitor was negligible. When the annulus was concentric this assumption was va l i d but one may question that this may not be the case especially when the annulus was at the maximum eccentricity, i . e . , the inner tube was touching the outer tube. An experimental check was therefore made for the position e = 1.0 by placing the thermocouple at three different points — a, b and c shown in Figure l6 below. At one particular flow condition the maximum variation in temperature between any of the above three points was found to be less than 3 psr cent. a FIG. 16. h3 The deflection or sag of the inner cylinder could cause an error in the transient cooling of the model. When the annulus was concentric any small sag in.the inner cylinder could not appreciably influence the flow pattern because of the re lat ive ly large annular thickness. Also the capacitor was mounted near the downstream support 'so.that any small sag was around the middle of the test section which has much less influence on the capacitor. For the eccentric annulus, however, the effect, of the sag', of the inner cylinder could be apprec-iable especially at higher eccentricity. To avoid this a hollow alum-i inium tube was used to support the capacitor i n the eccentric annulus. This is shown i n Figure 2. The tota l deflection in the centre of the tube was, in this case, l imited.to 0.010 i n . which is 10 per cent of the maximum annulus thickness. It was assumed that this tolerance was within the required accuracy of the experiment. The errors due to cal ibration of the recording instrument and also of the thermocouples were not considered to be of any con-sequence. The experiment was conducted i n a small temperature range and i t was.found that within this range the•temperature-emf response of a copper-constantan thermocouple was very close to l inear . There-fore, in the dimensionless temperature-time plot , Figure 15, i t was sufficient to plot the logarithm of the emf ratios d irect ly from.the recorder chart. Also because of the dimensionless temperature, the cal ibration of the recording instrument was not required as long as i t demonstrated a l inear scale. 1+1+ CHAPTER V CONCLUSIONS 1. The use of the transient test technique was found to be a simple and quick method for determining the heat transfer chara-ter i s t i c s in an annulus. Heat transfer coefficients in an eccentric annulus were found to decrease with increase i n eccentricity. The decrease of heat transfer coefficient did not seem to be l inear ly •related to the increase of eccentricity. However, the effect of eccentricity was more pronounced in the range 0 e <0.5 where the value of heat transfer coefficient decreased considerably. For example, at Reynolds number of 50,000 approximately 67 per cent of the to ta l decrease occured i n the range 0 ^ e <0.5 while the other 33 P e r cent occured in the range 0.5 ^ e ^1.0. Equations of Wiegand (2) and Monrad and Pelton (3) were found to correlate the annulus data of the present investigation very well . 2. Almost isothermal data was obtained by the transient method. In the annulus, the air temperature was pract ica l ly constant for one complete: run. and therefore the evaluation of the properties could be easi ly made at this temperature. This eliminates the trouble of determining the f l u i d f i lm temperature required in other methods of determining the heat transfer data. 3• The temperature of the cooling capacitor was measured in an arbitrary scale during the experiment. Therefore, careful 1+5 cal ibration of the thermocouple and the recording instrument was not required. RECOMMENDATION FOR FUTURE WORK The present investigation may be considered as an explor-atory work for more extensive studies in the same f i e l d . One of the important studies which is yet to be done is the behaviour of the heat transfer coefficients in the region where neither the thermal nor the hydrodynamic boundary layer has developed completely. This s i tua-tion may be encountered in practice i n a reactor where the heat trans-fer from the fuel rod is expected to take place from the very start of the tube. This investigation can probably be made by using the same technique provided a method is found to isolate the capacitors thermally along the longitudinal direct ion. BIBLIOGRAPHY London, A. L . , Nottage, H. B . , Boelter, L. . .M. K. "Determination of Unit Conductances for Heat and Mass Transfer by the Transient Method" Ind. and Eng. Chem. V o l . 33, 19^1, P- ^ 67 Wiegand, J . H. "Discussion on.Annular Heat Transfer Coefficients for Turbulent Flow by McMillen and Larson" Am. Inst. Chem. Engrs. Trans. V o l . i n , 191*5, p. U7 Monrad, C. C , Pelton, J . F . "Heat Transfer, by Convection in.Annular Spaces" Am. Inst. Chem. Engrs. Trans. V o l . 38, 19U2, p. 593 Foust, A . S . , Christ ian, G.A. "Non-boiling Heat Transfer in. Annuli" Am. Inst. Chem. Engrs. Trans. V o l . 30, 191+0, p. 5^1 Barrow, H. "Fluid Flow and Heat Transfer in an.Annulus with a Heated Core Tube" Proc. Inst. Mech. Engrs. V o l . 169, 1955, p. 1113 a Mueller, A. C. . '!Heat Transfer from Wires to Air i n Paral le l Flow " Am. Inst. Chem..' :Engrs. Trans. V o l . 38, 19^2, p. 613 Davis, E . S. "Heat Transfer and Pressure Drop in Annuli" Trans.•ASME V o l . 65, 19^3, P . 755 Knudsen, J . G . , Katz, D . . L . "Fluid Dynamics and Heat Transfer" McGraw-Hill Book Co. .Inc . 1958 Stein, R. P . , Begel, W. "Heat Transfer to Water in Turbulent Flow in Internally Heated Annuli" Am. Inst. Chem. Engrs. Jour:... V o l . k, 1958, p. 127 hi (10) Mizushina, T. "Analogy between F lu id Fr i c t i on and Heat Transfer in Annuli" General Discussion on Heat Transfer - ASME/IME 1951, P. 191 (11.) Barrow, H. . "A Semi-theoretical Solution of Asymmetric Heat Transfer in. Annular Flow Jour. Mech. Eng. Sc. V o l . 2 , i 9 6 0 , P . .331 (12) Deissler, R. G . , Taylor, M. F . "Analysis of Fu l ly Developed Turbulent Heat Transfer and Flow in.Annulus with Various Eccentric i t ies" NACA - TN 3U51, March, 1955 (13) Leung, E . Y . , Kays, W. M . , Reynolds, W. C. "Heat Transfer with Turbulent Flow in Concentric and Eccentric Annuli with Constant and Variable Heat Flux" NASA - N62 - 131V3, August, 1962 (lh) Eber, G. R. "Experimental Investigation of the Brake Temperature and Heat Transfer of Simple Bodies at Supersonic Speeds" Archiv. 6 6 / 5 7 , Peenemude, Nov., 19U1 (15) Fischer, W. W., Norris , R. H. "Supersonic Convective Heat Transfer Correlation from.Skin Temperature Measurements on a V -2 Rocket in Fl ight" Trans. ASME V o l . 71 , I9U9, p. ^57 (16) Garbett, C. R. 'The Transient .Method for Determining Heat Transfer Conductance from Bodies i n High Velocity . F lu id Flow" Heat Transfer and F lu id Mechanics Institute Preprints of Paper, 1951 Stanford Univ. Press (17) Kays, W. M . , London, A. L . , Lo, R.. K. "Heat Transfer and Fr i c t i on Characteristics for Gas Flow Normal to Tube Banks - Use of a Transient Test Technique" Trans. ASME V o l . 76, 195^, P- 387 (18) Kreith, F , "Principles of Heat Transfer" International Text Book Co. 1958 (19) M i l l e r , P . , Byrnes, J . J . , Benforado, D . M . "Heat Transfer to Water in an Annulus " Am. Inst. Chem.. Engrs. Jour. V o l . 1, 1955, P- 501 (20) ASME Power.Test Codes, Chap, 4, Part. 5 (21) Rothfus, R. R. "Velocity Distribution and F lu id Fr ic t ion in Concentric Annuli " D. Sc. Thesis, Carnegie Inst. Tech. 1948 (22) Stein, R. P . , Begell , Wm. "Heat Transfer to Water in Turbulent Flow in Internally Heated Annuli" Am. Inst. Chem. Engrs. Jour. V o l . 4, 1958, p. 127 (23) Giedt, W. H. "Principles of Engineering Heat Transfer" D. Van Wostrand Co. Inc. , 1958 (24) Jordan, H.P. "On the "Rate of Heat Transmission between F lu id and Metal Surfaces" Proc. Inst. Mech. Engrs. Parts 3 - 4 , 1909, p. 1317 (25) Foust, A. S. , Thompson, T. J . "Heat Transfer Coefficients in Glass Exchangers' Am. Inst. Chem. Engrs. Trans. V o l . 36, 1940, p. 555 (26) Zebran, A. H. 'Clar i f icat ion of Heat Transfer Characteristics of Fluids in Annular Passages" Ph. D. Thesis, Univ. of Michigan 1940 (27) McMillen, E . L . , . L a r s o n , R. E . "Annular Heat Transfer Coefficients for Turbulent Flow" Am. Inst. Chem. Engrs. Trans. V o l . 40, 1944, p. 177 h9 APPENDIX I AIR METERING SYSTEMS The a ir flow rate was measured to determine the Reynolds number of the flow. The following two methods were used: Method I . This method was used for Capacitor No. 1 at eccentricity, e = 0. The a ir flow rate was calculated based on a measurement of the maximum point velocity in the annulus. For laminar flow i t has been shown that the radius at the point of maximum velo-c i ty is related to the other r a d i i of a concentric annulus by the equation: r. m / 2 2 l o g e ( r 2 / r x ) =ex Rothfus (21) has shown experimentally that the point of maximum velocity for isothermal turbulent flow of a ir in a concentric annulus was the same as that for laminar flow such as calculated from equation [ l l ] . Later Knudsen and Katz (8) corroborated the same fact with their experimental results . It has also been reported by these authors that the turbulent flow velocity prof i le was very f la t in the v i c i n i t y of the point represented by equation [ l l j . Knudsen and Katz gave a relat ion of the average to the maximum velocity as ^avg/^max = ^ ^ T S with a diameter rat io of 3-6. In the present invest-igation, however, the diameter rat io was 3>0. For this rat io Denton* *Mech. Eng. Dept., Univ. of Br i t i sh Columbia, M.A.Sc. thesis in. preparation. 50 studied the velocity profi les in turbulent flow and concluded that O.876 was correct for the present annulus. The average flow velocity thus calculated had a maxiipum variation of 1.8 per cent. The maximum point velocity was measured by a pitot static tube placed at the radius, r m . It was mounted at the exit of the test sec-t ion and was connected to an accurate micro^manometer capable of read-ing low pressures from zero i n . of water head to 6.0 i n . of water head. The temperature of the a ir stream was measured by an ordinary mercury i n glass thermometer. The thermometer was graduated to 1 F . It was found that the a ir temperature did not change appreciably dur-ing this part-, of the experiment. The Reynolds number was f i n a l l y calculated from the above data. Method I I . This method was used i f o r the following cases: 1. Capacitor No. 1 at eccentricity, e = 0.25, 0.5 and,1.0. 2. Capacitor No. 2 at eccentricity, e = 0. The a ir flow rate was determined using a f la t plate or i f ice mounted on a 3 i n . schedule hO pipe. Figure 3 shows the position of the or i f ice with respect to the test section and fan posit ion. The or i f i ce was designed in accordance with ASME -.Power Test Code (20). Straight lengths of eleven equivalent diameters at the in let and seven equiva-lent diameters at the outlet of the or i f ice were provided as required by the code. The flow coefficient was taken from the Table given in the same reference. Table I gives the details of the o r i f i c e . 51 TABLE I. DETAILS OF ORIFICE Pipe diameter, (in.) 3.068 O r i f i c e diameter, (in.) 1.628 Flow c o e f f i c i e n t 0.635 Pressure taps Flange taps M a t e r i a l S t a i n l e s s S t e e l The o r i f i c e pressure drop and i n l e t side s t a t i c pressure of a i r were measured by two ordinary U-tube manometers f i l l e d with dyed water f o r be t t e r v i s i b i l i t y . The manometer scales were graduated i n tenth of an inch. Since the ultimate objective was to determine the flow Reynolds number, a small error i n the flow metering system was u n l i k e l y to cause any appreciable error i n the f i n a l p l o t of the r e s u l t s . A i r temperature was measured by a mercury i n glass thermo-meter graduated to 1 F. 52 APPENDIX II PHYSICAL PROPERTIES AND CONSTANTS OF THE THERMAL CAPACITORS The dimensional details of the two thermal capacitors are shown i n Figure 3» The physical properties and constants of the cap-acitors as required for the calculation of the heat transfer coeffits:" cdentBc are given i n the following table. TABLE II. PROPERTIES OF THE THERMAL CAPACITORS Capacitor No. 1 Capacitor No. 2 Material Length, (in.) piameter, Dj (in.) Mass, (gm.) Cl (Btu/lb F) Kg, (Btu/hr F ft) 99-9$ Copper 2.1250 1.0 103.2312 0.0915 223.0 99.9$ Copper 1.4703 1.0 83.8931 O.O915 223.0 The calculated values of Blot number based on an average heat 2 transfer coefficient of 15 Btu/hr f t F are given i n the following table. TABLE III. BIOT NUMBER OF THE CAPACITORS Capacitor No. 1 Capacitor No. 2 N B i 0.00583 0.00068 53 APPENDIX III PHYSICAL PROPERTIES OF AIR The properties of a ir required for the calculation of Reynolds and Nusselt numbers were the following: 1. Density, ^ 2. Dynamic v i s c o s i t y , ^ 3. Thermal conductivity, K The density of a ir was determined from the equation of state. The viscosi ty and the thermal conductivity of a ir were taken from the tables given by Giedt (23). In Figures L7 and 18, these properties are shown as a function of temperature only at ih.'J lb/in^abs. pressure. 0.047 0 . 0 4 2 • " 1 1 1 ' 1 1 » 60 70 80 90 100 TEMPERATURE, (p) PIG. 17. DYNAMIC VISCOSITY OP AIR AT ATMOSPHERIC PRESSURE 0.019 0.018 0.017 0.016 0.015 0.014 60 70 80 90 100 TEMPERATURE, (?) PIG. 18. THERMAL CONDUCTIVITY OP AIR AT ATMOSPHERIC PRESSURE \J1 APPENDIX IV SAMPLE CALCULATIONS The following calculations of the heat transfer coefficient. was done for Run 10 of the concentric annulus test. From the experi-mental data provided in Table.IV, the values of T* were computed which are given below. e . (sec) (Dimensionless) 0 1.0 10 0.925 20 0.860 30 0.797 ^0 O.7J+I 50 0.693 6o 0.6UU 70 0.603 The above values are plotted in Figure 15 which gave a straight l ine . The slope of this l ine was found to be 26 .2 hr 1 . It was Slope was given by, shown in Chapter II that, = — . Therefore, the heat transfer coefficient C ' "-W ££0 - *-Tl BtuAr f t 2 P. APPENDIX V TABLES OF EXPERIMENTAL RESULTS 58 TABLE IV. TEMPERATURE-TIME DATA FOR CAPACITOR NO. 1, e.= 0. (VALUES OF T AND T a ARE IN CHART DIV.) Run Time, 9 (sec) 0 10 20 30 40 50 60 70 .1 90.1 87.7 85.I 83.0 81.1 79-3 77-7 76.0 4i.0 2 88.8 86.0 83.5 81.1 79-0 77-0 75.2 73-4 40.0 3 86.5 83.2 80.5 77-8 75.5 73.2 71.3 69.6 39-1 4 84.0 80.6 77-6 74.7 72.2 70.0 67.8 65.9 38.5 5 83.6 80.0 76.8 74.0 71.2 68.8 66.7 64.8 38.2 6 82.7 79-1 76.0 73-1 70.4 68.0 65.9 64.0 38.0 7 83.8 80.1 76.8 73-8 71.0 68.6 66.3 64.2 37-6 .8 84.0. 79.8 76.O 72.8 70.0 67.2 64.9 62.7 37.4 9 84.8 . 80.3 76.5 73.0 70.0 67.3 65.0 62.7 37-4 10 84.0 79.4 75.3 71.7 68.6 65.7 63.3 6 l . l 37-2 11 85.8 81.0 76.6 72.8 69.5 66.5 63.8 61.6 37-0 12 82.8 78.2 74.1 70.6 67.5 64.6 62.1 60.0 36.7 13 85.4 80.2 7-5.6 71.6 68.0 65.O 62.2 59-8 36.0 59 TABLE V. DATA FOR COMPUTING REYNOLDS AND NUSSELT NUMBERS FOR CAPACITOR NO. 1, e = 0 . Run T a e %e A h Nu (F) ( in water) ( i b / f t 3 ) ( l /hr) (Btu/hrft 2 F) 1 79-0 0 . 0 8 .0747 16,700 17.6 7.91 88 .0 •2 78 .0 0.128 1.0746 21,100 19.8 8 .90 99-0 3 76.5 0.195 .071+6 26,100 23 .5 10.55 117.3 4 76 .0 O.285 .07U3 31,500 26.2 11-77 131.0 5 75-7 0.335 .071+5 34,200 28 .0 12.58 l 4 0 . 0 6 75:-5 0 . 3 9 0 ..0746 36,900 28 .2 12:..68 11+1.5 7 74.8 0.1+50 . 0 ^ 5 39,600 28.6 12.85 1^3.5 8 74.6 0.520 .Ojhh 1+2,700 31.7 14.23 159.2 9 7^.6 O.565 .07I+I+ 44,500 32.6 11+.65 164.0 1 0 74.5 0.630 1+6,900 35.3 15.85 177-2 l l 74 .0 0.705 .071+1+ 1+9,700 35-8 16.09 180.0 12 73-4 0 . ' 7 l a .071+8 51,250 •35-9 16.10 180.4 13 73 .0 • 0 . 818 .0747 53,750 38.1 17.10 191.8 TABLE VI. TEMPEMTURE-TIME DATA. FOR CAPACITOR WO. 1, e = 0.25 (VALUES OF T AND T a ARE IN CHART DIV.) Run Time, G (sec) 0 10 •20 30 40 50 60 70 14 . 82.5 80.3 78.3 76.6 74.8 73-2 71.7 70.3 34.0 15 81.8 . 8 0 . 0 78.5 76.9 75-5 74.3 73.0 71.8 47.0 16 83.1 8 0 . 7 78.6 76.7 74.8 73.0 71.6 7 0 . 0 •38.0 17 81.6 79-5 77-5 75.5 73.8 72.2 70.7 69.3 4o.o 18 82.8 80.4 78.2 76 .1 74.3 72.7 71.0 6 9 . 8 4i.o 19 8 0 . 0 78 .2 76.2 74.7 73-1 71.7 70.3 6 9 . 2 48.5 20 8 1 . 8 79-7 77.6 75-7 74.0 72.3 71.0 6 9 . 7 4 7 . 0 21 79.3 77-0 75-0 73.2 71.6 7 0 . 0 68.6 67.2 46.5 22 8 1 . 9 79.3 77-0 75-0 73.0 71.3 6 9 . 8 6 8 . 3 48.0 23 7 8 . 0 75-4 73.2 71.1 69.3 67.7 6 6 . 1 64.8 46.5 2h 80.1 77.4 75.0 72.9 70.9 69.O 67.5 66.0 48.0 25 : 80.3 77.7 75.2 73-0 71.0 6 9 . 2 67 .7 66.25 48.0 26 79.2 76.6 74". l 72.0 7 0 . 0 68.2 66.7 6 5 . 2 48.5 27 82.0 79.0 76.1 73-7 71-5 69.6 67.8 66.2 48.2 28 81.2 7 8 . 0 75-2 72.8 70.6 68.6 66.8 6 5 . 1 48.0 29.. • 80.6 77.1 74.1 71.5 6 9 . 2 6 7 . 1 65.3 63.7 47.3 TABLE VII. DATA FOR COMPUTING REYNOLDS AND NUSSELT NUMBERS FOR CAPACITOR NO. 1, e = 0.25 Run T a h w h s e %e A h Nu • (F) (in.water) (in.water) (ib/fto (l/hr) (Btu/hrft 2 F) l 4 73 .5 1.35 1.1 .0747 16,430 15.00 6.73 75.4 15 85-5 ."'.1.70 1.5 .0729 17,920 17.60 7.92 86 .6 16 75-5 2 .10 1.8 .0743 20,400 17.90 8.06 90.0 17 78 .0 2 .60 2 .3 .0740 22,600 18.20 8.18 ' 9 1 . 0 18 79-0 3-20 2 .8 .0742 25,050 19.82 8.92 99 .0 19 87 .O 3.90 3.4 .0730 27,080 21.90 9.85 107.5 20 8 6 . 0 4 .90 4 .3 .0735 30,550 22.32 10.05 114.6 21 85 .5 6.85 6 .2 .0738 36,200 23.94 10.77 118.0 22 87 .O 7.90 ' .7 .0 .0737 38,800 26.50 11.92 130.2 23 85 .O 9.70 8 ,7 .0742 43,300 28.20 12.69 139.0 2k 86 .5 11.30 10 .1 .0746 46,750 30.10 13.54 148.2 25 86 .5 12.40 11 .1 .0747 49,060 29.80 13.40 146.7 26 87 .O 13.40 12 .0 .0747 50,800 31.60 14.21 155.2 27 86 .5 14.45 12.9 .07-50 53,ooo 32.40 14.59 159.8 28. 86 .0 15.65 14.1 .0751 55,200 34.30 15.42 168.6 29 85.6 16.80 15 .0 .0756 57,4oo 36.80 16.55 181.0 62 TABLE VIII. TEMPERATURE-TIME DATA FOR CAPACITOR NO. 1, e = O.50 (VALUES OF T AND T a ARE IN CHART DIV.) Run Time, Q (sec) T a 0 10 20 30 40 . 50 60 70 3.0 86.0 84.3 82.9 81.5 80.0 78.8 77.7 76.6 47.0 31 85.0 83.1 81.5 80.0 78.5 77.1 75-9 74.8 46.0 32 86.5 84.7 82.9 81.3 80.0 78.7 77.5 76.3 48.5 33 82.2 80.5 79-0 77-6 76.I 74.8 73.7 72.6 48.6 34 82.1 80.3 78.7 77.1' 75.7 74.4 73.1 72.0 49.0 35 85.0 83.O 81.2 79-5 77.8 76.5 75-2 74.0 48.0 36 80.8 79-0 77.1 75-5 74.0 72.6 71.4 70.2 48.0 37 86.1 83.6 81.4 79.5 77.5 76.O 74.4 73.0 48.5 38 80.8 78.7 76.7 74.9 73.2 71.7 70.3 68.9 47.0 39 79-8 77-3 75-2 73.3 71.6 70.0 68.6 67.3 45.0 1+0 83.2 80.5 78.1 76.O 74.0 72.2 70.5 69.0 48.0 4i 84.4 ' 81.7 79-2 77-0 75.0 73.3 71-7 70.2 48.5 42 79-0 76.4 74.0 72.0 70.1 68.6 67.0 65.7 47.0 43 81.2 78.2 75.2 73-0 70.8 68.8 67.2 65.7 46.0 hh 81.0 77.8 74.8 72.3 70.0 67.9 66.1 64.5 45.0 TABLE IX. DATA FOR COMPUTING REYNOLDS AND NUSSELT NUMBERS FOR JCXPACITOR NO. 1, e = 0 . 5 0 . Run e WRe A h %u (F) (in.water) (in.water) ( l b / f t 3 ) ( l /hr) (Btu/hrft 2 F) 30 86 .2 1-5 1.25 .0728 16,580 14.17 6.37 69 .5 31 8 6 . 0 2 .1 1.90 .0730 19,620 15.81 7.12 76 .7 32 87 .0 2 .5 2 .20 .0730 21,700 16.20 7.30 79-6 33 88 .0 3.2 .2 .80 .0728 24,100 17.25 7.76 84 .6 34 88 .0 4 .0 3 .60 .0728 27,000 18.95 8.52 92 .9 35 8 7 . I 4.4 3 .90 .0730 28,800 19.10 8.60 93-7 36 87 .O 5-5 - 4 . 9 0 .0734 31,800 20.2 9.08 98.25 37 8 6 . 5 6 .7 6 .00 .0738 • 35,800 22.4 10.08 110.0 38 86 .0 6 .8 6 .00 .0737 35,600 22 .4 10.08 110.3 39 84 .0 8 .25 7 .20 .0743 39,700 23.7 IO.65 117.0 ko 8 6 . 0 10 .0 8 .80 .0740 43,150 26 .7 11.0 120.2 kl 86 .0 11.9 10.70 .0747 48,000 26.22 11.80 129.0 kl 85 .O 14.0 12.30 .0750 5 l , 4oo 28.9 13.00 142.3 43 84.0: 16.2 14.30 .0758 56,750 30.4 13.65 150.0 kk 8 3 . 0 18.15 16.05 .0758 59,900 32.25 14.50 159.5 TABLE X. TEMPERATURE-TIME DATA FOR CAPACITOR N 0 . . 1 , e = 1 . 0 (VALUES OF T. AND T a ARE IN CHART DIV.) Run Time, 0 (sec) T a 0 10 20 30 40 50 60 70. 45 85.2 84.0 82.9 81.9 80.8 79-9 79-0 78.8 48.7 46 80.4 78.9 77-5 76.1 74.8 73.7 72.6 71.6 38.0 47 82.2 80.7 79.3 78.0 76.7 75-5 74.5 73-5 42.0 48 78.5 77-1 75-9 74.7 73-7 72.6 71.7 70.8 45.5 49 79.7 78.3 77.0 75.8 74.8 73.8 72.6 71.8 47.O 50 80.8 79.4 78.0 76.8 75.5 74.5 73.4 72.5 48.0 51 82.5 80.8 79.3 77-9 76.5 75-3 74.2 73-0 48.3 52 85.9 83.8. 8I.9 80.0 78.5 76.9 75-5 74.2 47.5 53 84.5 82.1 80.0 78.2 76.5 75.0 73-6 72.2 47.0 54 . 85.O 82.7 8O.5 78.3 76.5 74.8 73.3 71.9 46.7 55 82.8 8O.5 78.3 76.4 74.7 73-0 71.6 70.2 46.5 56. 81.0 78.4 76.2 74.2 72.3 70.7 69.2 67.8 45.5 57 84.0 80.8 78.0 75-7 73.6 71.6 69.8 68.1 44.7 58 83.2 80.1 77.4 74.9 72.7 70.7 68.9 67-2 42.5 TABLE XI. DATA FOR COMPUTING REYNOLDS AND NUSSELT NUMBERS FOR CAPACITOR NO. 1, e = 1.0. Run T a e NRe A h WNu (F) (in.water) (in.water) ( i b / f t 3 ) (1/hr) (Btu/hrft^) 45 88.0 1.3 1.1 .0725 15,600 11.49 5.17 56.25 46 76.0 1.3 1.2 .0743 16,070 12.25 5.51 61.50 47 79.0 1-9 1.1 .0738 19,300 21,450 12.80 5.75 63.80 48 84.0 2.4 2.2 .0734 14.13 6.35 69.70 49 85.0 3*2 2.8 .0733 24,600 15.35 6.90 75.60 50 87.0 -3-7 3-2 .0730 26,350 15.45 6.95 76.OO 51 87.0 4.7 4.1 .0730 29,700 16.90 7.60 83.00 52 86.0 6.2 5.3 .0736 34,400 19.26 8.66 94.50 53 86.5 8.4 7.2 .0740 40,200 20.00 9.00 :98'.4o 54 85.0 10.2 8.8 '.0742 44,300 47,800 22.10 9.94 108.90 55 85.0 11.8 10.2 .0747 22.15 9.95 109.00 56 83.5 13.6 11.7 .0750 51,600 24.60 11.05 121.50 57 82.0 16.05 13.8 .0755 56,200 26.22 11.80 130.00 58 80.0 18.35 15.9 .0764 65,000 26.30 11.80 130.50 66 TABLE XII. TEMPERATURE-TIME DATA FOR CAPACITOR NO. 2, e = 0 (VALUES OF T AND T a ARE IN CHART DIV.) Run Time, 9 (sec) T a 0 10 20 30 ko 50 60 70 59 86.6 85.O 83.5 82.0 80.7 79-3 78.2 77-0 1+7.0 60 79-7 78.0 76.5 75-2 7U.0 72.8 71.7 70.6 ^7.5 6 i 80.0 77-9 76.1 74.5 73.0 71.6 70.2 69.0 1+6.7 62 79-8 77.5 75-4 73-6 71.8 70.2 68.9 67.5 U6.0 8l.O 78.0 75.5 73-3 71.2 69.5 67.7 66.3 1+5.0 6k 79-7 76.7 73-8 71-5 69.3 6l.k 65.7 6k.2 kk. 0 65 80.2 76.5 73-3 70.4 67.9 65.7 63.7 62.0 1+3.0 TABLE XIII. DATA FOR COMPUTING REYNOLDS AND NUSSELT NUMBERS FOR CAPACITOR NO..2, e = 0 Run T a K h s e % e A . h % u (F) (in.water) (in.water) ( l b / f t 3 ) (T/ihr) (Btu/hrft 2F) 59 86.0 1.25 1.1 .073 15 , 4 0 0 14.35 7.57 83.0 6o 86.0 2.90 2-5 .0732 23,420 17.10 9.03 99.0 61 85.O U.70 4.0 .0735 29,900 21.10 11.15 122.0 62 8U.5 7.05 6.1 .0740 36,900 23.35 12.32 135.0 63 83.O 10.il 9.0 .07U7 ^5,000 27.30 ih.ko 158.4 6h 82.0 13.2 11.k .0752 51,000 30.30 16.00 176.2 65 80.5 18.35 16.0 .0745 59,900 34.60 18.28 202.0 

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