54 HT T LIDELARVEVAREA 3 4456 03L0LOS 1 ORNL-1370 Physics TURBULENT FORCED CONVECTION * HEAT TRANSFER IN CIRCULAR TUBES CONTAINING MOLTEN SODIUM HYDROXIDE CENTRAL RESEARCH LIBRARY DOCUMENT COLLECTION LIBRARY LOAN COPY DO NOT TRANSFER TO ANOTHER PERSON If you wish someone else to see this document, send in name with document and the library will arrange a loan. OAK RIDGE NATIONAL LABORATORY OPERATED BY CARBIDE AND CARBON CHEMICALS COMPANY A DIVISION OF UNION CARBIDE AND CARBON CORPORATION (=4 POST OFFICE BOX P OAK RIDGE. TENNESSEE UNCLASSIFIED - ORNL-1570 This document consists of 42 pages. Copy R of 384 copies, Series A. Contract No. W-T405, eng 26 Reactor Experimental Engineering Division TURBULENT FORCED CORVECTION HEAT TRANSFER IN CIRCULAR TUBES CONTAINING MOLTEN SODIUM HYDROXIDE H. W. Hoffman Date Issued: OAK RIDGE NATIONAL LABORATORY Operated by CARBIDE AND CARBON CHEMICALS COMPANY A Division of Union Carbide and Carbon Corporation Post Office Box P Oak Ridge, Tennessee MK UNCLASSIFIED _2- ORNL 1370 Physics INTERNAL DISTRIBUTION 1. G. T, Felbeck (C&CCC) 33, J. 5. Felton 61. W. D. Powers 2-3., Chemistry Library 34. A, S. Householder 62. R. F. Redmond L. Physies Library 35. C. S, Harrill 63. W, B. Harrison 5. Health Physics Library 36, D. S. Billington 64. W, S, Farmer 6. Biology Library 37. D. W. Cardwell 65. L. D, Palmer 7. Metallurgy lerary 38. E. M, King 66. L. Cooper 8-9. Training School - “§ 39. R. N, Lyon 67. P, C. Zmola 10. Reactor Experlmental - h0, J. H. Buek 68. M. Richardson Engineering Library 4L1. R. B. Briggs 69. F. E. Lynch 11-14. Central Files 422, A. S, Kitzes 70. M. Tobias 15. C. E. Center 43. O, Sisman 71. G. A, Cristy 16. W, B. Humes (K-25) Lly. R. W. Stoughton 72. W. K. Ergen 17. L. B, Emlet (Y-12) L5. C. B, Graham 73. C. P. Coughlen 18, C. 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Bacher, California Institute of Technology lBO-Béhf Given distribution as shown in TID-4500 under Physics Category DISTRIBUTION PAGE TO BE REMOVED IF REPORT IS GIVEN PUBLIC DISTRIBUTION ABSTRACT An experimentel determination has been made of the heat transfer coefficients for molten sodium hydroxide flowing in turbulent forced convection through a tube of circular cross section and a length to diameter ratio of 200. Heat transfer coefficients in the region of fully developed turbulent flow are reported for the Reynolds modulus rasnge of 6000-12000 and the temperature range of 700-900°F. The eguation, Nu/Pro'h = 0.021 Re°'8, was found to correlate the data for (L/D) values above 100. Thermal entry lengbths are calculated for each run. Results show that molten sodium hydroxide may be considered an ordinary fluid - i.e., any fluid other then the liquid metals - as far as heat transfer is conecerned. II. III. Iv. TABLE OF CONTENTS INTRODUCTION EXPERIMENTAL WORK A. Description of Apparatus B. System Calibration C. Method of Operation RESULTS A. Method of Calculation B. Correlation of Heat Transfer Coefficient C. Thermal Entrance Region D. Analysis of Errors NOMENCIATURE BIBLIOGRAPHY APPENDIX I. Sample Calculation APPENDIX II. Physical Properties APPERNDIX III. Solution of Conduction Equation for Tube with Heat Generation in Wall t:~a ~ w | o [1] 18 21 21 2L, 29 32 33 35 36 38 L0 INTRODUCTION Sodium hydroxide can be used as a high temperature heat transfer medium. This fluid freezes et 604°F and possesses temperature stability to at least 1200°F. No data on the heat transfer properties of molten sodium.hydroxide are available in the technical literature. Therefore, an experimental in- vestigation was underteken to determine the convective heat transfer coefficient for this fluid in turbulent flow within tubes of circular cross section. In designing heat transfer equipment it is necessary to know the conductance, or coefficient of heat transfer, between a surface and the fluid flowing past the surface so as to be able to estimate the amount of heat transferred or to prediet the temperature differences existing within the system. This coefficient is defined by the equation ho A (1) where (q/A) is the heat flux through the metal-fluid interface in Btu/hr-£t°, tg, the temperature of the metal surface at this interface and t,, the fluid mixed-mean tempersture. A number of investigatorsh’s’a* have developed systems, both of the double-tube exchanger type and the single-tube electrically hested wall type, which enable determination of the temperature, tg. For reasons of simplicity of assembly and more accurate determination of the heat input and the heat transfer area, the electirically heated test section was chosen for this experiment. *Nunbers refer to references to the literature glven in section v e i e bt e 6 Several factors mist be considered in designing this type of test unit for use with molten sodium hydroxide. 1. Corrosion. The extreme corrosiveness of molten sodium hydroxide drastically restricts the materials available for constructing the system. At the time the problem was begun, static corrosion test32 indicated that only three construction materlials would be acceptable for a system containing sodium hydroxide. These were pure silver, graphite and A-nickel. Of these silver appeared to be the best material. However, silver is structurally weak and at high tempera- tures is unable to support its own weight. To be effective corrosion- wise the silver must be oxygen free and thus the cost of a silver- lined system becomes prohibitive. Graphite would require heavy, awkward structural sections to give sufficient strength. Hence, nickel remeins as the only feasible construction material. While weeker than most construction metals it still possesses reasonable strength at.high temperatures and is commercially available in most of the required forms. Electrical Conductivity. While the electrical conductivity of molten sodium hydroxide is not high as compared to the liquid metals, 1t is still sufficlently large to necessitate that the system be designed to minimize.the heat generation in the test fluid. From this aspect nieckel was also satisfactory since its electrical conductivity was such as to allow 95% of the heat generation to occur in the tube well for a reasonably sized experimental system. 3. Pumps. Due to the unavailability of pumps for handling molten sodium hydroxide another method of causing fluid flow through the system must be devised. This was aecomplished by pushing the fluid through the system under the pressure of an inert gas, argon . 4. Melting Point. The high melting point of sodium hydroxide (604°F) increases the difficulty of system design. II. EXPERIMENTAL WORK A. Description of Apparastus The experimental system designed to measure the heat transfer coefficient is illustrated schematically in Figure 1. It consisted of a sump tank, a test section with associated power supply and temperature and power indieating de;iees, a tank resting on a scale for measuring the fluid flow rate and a gas system for moving the fluid through the test section. Figures 2, 3, and 4 present several views of the apparatus. The sump and weigh tanks were electrically heated by strip heaters, and the lines connecting the test section to the tanks were traced with Calrod heaters. The entire system was well-lagged and was maintained at a temperature of 650°F. Preliminary tests indicated that even when the weigh tank was rigidly connected to the rest of the system the scale readings were consistent and accurate. This was checked by visually observing the scale action while introducing weighed amounts of water into the tank and while placing weighed lead bricks on the scale. It was possible to reed the weight of fluid entering or leaving the weigh tank to within 1/2 pound. UNCLASSIFIED DWG. 16324 THERMOCOUPLES VY \/ ©® 00000000000000Q (0060000 0005000) | l | | l POWER SUPPLY | | | l— ————————— GAS SUPPLY b—m—— — — — — —I Test Section: Nickel Tube-3/16"0.D. X 0.035" Wall Thickness X 24" Long. 22 Thermocouples welded to outside tube wall. Fig. l. Schematic Representation of System for Measuring Heat Transfer Coefficients of Molten Sodium Hydroxide. UNCLASSIFIED Y-12 PHOTD 65227 T TL AR d! ekl FIGURE 2. VIEW OF PANEL BOARD S S T e IS UNCUASSIFIED P - E s Rl v - 17 PHOTO 65225 : , f = ' f a5 3 ] ; ..fi i} FIGURE 3. GENERAL VIEW OF EXPERIMENTAL SYSTEM — Ok — LN (Ia.a Hli‘n Y. 12 PHOTO 63226 . POWER LEADS . Tir’- VOLTAGE TAPS:;> ! | Wfi = 1."--‘, e MIXING POT R . TEST SECTION 9 TuBE WALL e e “THERMOCOUPLES S - ! - g ¥ - . i s - - - e i .,.H T FIGURE 4 TEST UNIT 12 In the design of an electrically heated test section, the size of the tubing used is dictated by severai factors; namely, allowable pressure drop through the test seetion, desired length to diameter ratio, avail- ability of the proper size tubing, the temperature drop through the tube wall as related to the temperature difference between the inside tube surface and the fluid mixed-mean, the physical size of the system as related to the fluid inventory, the minimization of heat generation in the test fluid and the total power available. On the basis of these considerations a test section was designed of nickel tubing, 3/16" 0.D. X 0.035" wall thickness, 24 inches long. With this test section & Plow at a Reynolds modulus of 10h could be obtained with a pressure drop of 35 psi. The heat generated in the tube wall accounted for 95% of the total heat generation. Since gas pressure was used to move the molten sodium hydroxide through the test seetion, the system.wfis capable only of intermittent operation with the fluid Fflowing first from the sump to the weigh tank and then from the weigh to the sump tank. A total of 220 pounds of sodium hydroxide was put into the system. The power was supplied to the test section by a transformer capable of delivering a maximum of 360 amperes at 7.5 volts. This was introduced to the test section through small rectangular copper flanges silver-soldered to the tube at the inlet and outlet of the test seetion. The current to the test section was measured with a multi-range ammeter. The accuracy of this meter is claimed as 3/4%. The temperature of the outer surface of the test section was measured at 20 points along the tube and at the power flanges at the inlet and outlet of the test section. The couples were of 36 gage chromel and alumel wire 15 and were welded to the tube surface by a single-pulse resistance welder. They were then wrapped around the tube for about one-quarter of a turn in order to minimize conduction losses along the thermocouple leads. At the two ends of the test section thermocouples were encased in pieces of two-hole ceramie insulator, inserted in small holes in the power flanges and arc-welded to the tube wall. All couples were then connected to a Leeds and Northrup portable precision potentiometer through selector switches. A commofi cold Junction was used. This consisted of an ice-water mixture in a one pint Dewar flask which was Jacketed by a can filled with thermal insulation. Temperatures for control purposes were measured by thermocouples of 28 gage iron and constantan wires located along the tubing and in thermo- wells in the tanks. The temperatures were indicated on a precision temperature indicator. The voltage was measured at the 10 locations along the test section indicated in Figure 5. Two of these voltage taps were loceted on the power flanges to the test section. The voltage taps were of 28 gage bare copper wire, arc-welded to the tube. These were then ecoated with #7 Sauereisen cement to reduce oxidation of the copper at the operating temperature of the system. The volltage was measured by a multiple-range Ballantine electronic voltmeter, for which an accuracy of 2% is claimed. The mixed-mean temperature of the fluid entering and leaving the test section was obtained by inserting mixing pots in the test line. These were located one inch from the inlet and outlet of the test section. The mixing pot was & two inch length of standard one inch nickel pipe capped at both ends. A perforated nickel disc was located at the ecenter. The inlet and outlet were tangentially positioned at the top and bottom. Volts UNCLASSIFIED DWG. 16325 _” AT o 5 / 4 ' > 3 / 2 / s /, | 7 // // O 0 2 4 6 8 i0 12 14 16 18 20 22 Distance from Entrance of Test Section, inches Fig. 5. Voltage Impressed on Test Section as a Function of Tube Length. 15 Thermowells were placed immediately behind the inlet and before the outlet. A mixing pot is shown in Figure 4. It is believed that with the whirling flow caused by the tangential inlet and the mixing caused by the diffuser dise, & good mixed-mean temperature was obtained. An aixalysis of the error due to conduction losses by the thermowell at an average fluid temperature of 950°F shows the correction to the observed thermoeouple reading to be negligibly smell. The mixing pots were - wrapped with a double layer of monel-sheathed nichrome heating wire. The current ‘through these heaters was adjusted to maintain a temperature of approximately 650°F in the mixing pot so as to prevent freezing of the sodium hydroxide and reduce heat loss from the fluid. The entire test unit - test section plus mixing pots - was Jacketed by a metal container filled ?i'l;h thermal insulation. System Calibration 1. Heat Loss In order to obtain an estimate of the heat loss from the system, current was passed through the tube wall without fluid flow through the test section. At equilibrium, the power input for a given average tube wall temperature with no fluid flow was taken to be the system heat loss for the same average tube wall temperature with fluid flow. The outside tube wall temperature is shown in Figure 6 as a function of the distance along the test section for various power levels. Figure 7 shows the system heat loss as a function of At, where At ris the average outside tube wall temperature, by gyve, minus the temperature of the system environment, te. The average outside tube wall temperature was cbtained by graphical integration of the curves of Figure 6. 1200 10O 1000 900 800 ELECTRICAL HEAT INPUT, Btu/hr: o — 33} A— 310 0— 269 v— 214 ¢— |98 Outside Wall Temperature, °F fl o 600 500 400 0 2 4 6 8 10 12 14 16 Distance from Entrance of Test Section, inches UNCLASSIFIED DWG. 16326 18 20 22 24 Fig. 6. Axial Profiles of Outside Tube Wall Temperature for Varying Heat Input with no Fiuid Flow. UNCLASSIFIED DwG. 16327 350 3 300 /( A 250 7 o £ . 3200 lva) - S’ L 150 / 100 /// 50 /,l // O« / 0 100 200 300 400 500 %OO 700 800 900 1000 (fw ave "fe ’ °F , . Fig. 7 Heat Loss from Test Section 18 2. Thermocouple Calibrafion Bead thermocouples of 36 gage chromel and alumel wire were calibrated at the melting points of lead and zine. OSamples of the wire used for the tube wall thermocouples were calibrated in a tube furnace. The results of these calibrations are in- dicated in Figure 8. A maximum correction of 5°F at 900°F was observed. 3. Effect of Tube Current on Thermocouples The possible effect of the current flowing through the tube wall on the readings of the thermocouples attached to the wall was considered. Therefore, with no flow through the tube, the system was allowed to reach equilibrium and the reading of one wall thermo- couple recorded. The power to the tube was then turned off and readings of this thermocouple taken every 15 seconds. These power- off readings were then extrapolated back to zero time. As is seen from Figure 9, the current in the tube wall has no effect on the readings of the thermocouples. Method of Operation Prior to operation, the entire system - sump and weigh tanks and the test unit - was heated to an average temperature of 650°F. During a sequence of runs this temperature would drift up to about 750°F due to the heat put into the fluid by the test section. The desired flow rate was obtained by proper adjustment of the gas pressures at the sump and weigh tanks. Runs lasted from 20 to 40 minutes depending on the fluid flow rate. Tt was found that during this period the system reached approximate equilibrium. This is indicated in Figure 10 which gives the readings 2 O 0 1., millivolts Test Thermocouple Reading, 5 o o @ ~1G- UNCLASSIKIRD DWG. 16328 *c = tfr O— BEAD THERMOCOUPLES CALIBRATED AT MELTING POINTS OF Pb & Zn x— TUBE WALL THERMOCOUPLES CALIBRATED IN TUBE FURNACE 4 6 8 10 12 14 16 True Reading, t,, millivolts Fig. 8. Calibration of Thermocouples 18 UNCLASSIFIED DWG. 16329 @® —-THERMOCOUPLE READING WITH CURRENT ON, SYSTEM AT THERMAL EQUILIBRIUM, o O O —THERMOCOUPLE READING WITH CURRENT OFF. o O g, mitlivolts » o » o -0z~ Themocouple Readin N i o O 2.5 2.0 30 45 60 . 75 90 105 120 135 Time, Seconds Fig.9. Effect of Current on Tube Wall Temperatures 150 21 from a given thermoecouple as a function of time during an experimental run. The tube wall thermocouples were read every 3 to 5 minutes during a run. Data were obtalned with the fluid flowing from the weigh to the sump tank, as well as from the sump to the weigh tank. The sodium hydroxide charged to the system had an original assay of 98.5% NaOH, 0.31% NapCO3 and the rest water. The water was removed during the melting cyele by maintaining the systeém under vacuun. ITI. RESULTS A. Method of Caleulation The coefficlent of heat transfer at any point, x, along the test section is known as the local coefficient of heat transfer and is defined | by the equation (tS = EE x . For values of x large enough to be beyond the thermal entrance region, hx = hx reaches a limiting value, h, the coefficient of heat transfer for the region in which turbulent flow has been fully established both thermally and hydrodynamieally. In order to determine the local coefficient of heat transfer, it is necessary to obtain from the experimental data the heat flux, the tempera- ture of the inside wall of the tube and the fluid mixed-mean temperature at each point along the tube. The inside tube wall temperature is calculated from the measured outside tube wall temperature by the equation 2 where the subscripts w and s indicate the outside and inside tube walls, . 2_p 2 ty-ts = 2#;1.(EW2 In Ty/Ts - Ta 75 ) (3) respectively, and r is the tube radius; ky; is the thermal conduetivity of the nickel tube. UNCLASSIFIED DWG. 16330 20 33 ___-ii—-——J"" Qo Z ' 'E // o - c ol9 o Q o Q a =3 Q O O e Py £ }.._ 18 17 4 6 8 10 12 i4 16 18 20 22 Time, minutes Fig. 10. Variation with Time of Thermocouple 10 During Run 12 Showing Approach to Equilibrium. 25 This equation takes intc account the effect of the heat generation in the tube wall on the temperature drop through the wall. The derivation of this equation is given irn Appendix III. The source term, W, is given by w= & 3" I (4) vhere E and I are the voltage and current at the test section, ¢ is the conversion factor, 3.413 Btu/hr-watt, and V is the volume of metal in ‘the tube wall of the test section. It is assumed that the heat generation is uniform in the tube wall. This is borne out by Figure 5 which shows the voltage impressed on the test section as a function of distance along the tube for a typical run. With uniform heat generation, the fluid mixed-mean temperature as a function of the distance down the tube is given by a straight line drawn between tp j and *m,o, the fluid mixed-mean inlet and outlet temperatures, respeetively. Actually, dve to heat losses in the wvicinity of the power flanges, not all of the heat generated is transferred into the test fluid. This results in & slightly lower slope for the mixed- mean temperature curve in the immediate vieinity of the power flanges and a correspondingly slight raising of the slope tkrough the central portion. However, since the heat loss through the power flanges was found to be low, less thar 2% of the total heat input, the error intro- duced by assuming a straight line between ty ; and ty o is small. The total heat generation in the tube wall is determined from the measured voltage drop across the test section and 4the current passing through the tube wall. Part of the total heat goes to external loss and the rest intc heating the fluid in the test seection. The external 24 heat loss can be obtained from Figure 7 for the average outside tube wall temperature for the given set of experimental data. The heat transferred into the fluid in passing through the test section is given by the equation ap = W ep(tm,0 - tm,1) (5) where w is the fluid flow rete in 1bs/hr, and tp,o and tm,i the fluid mixed-mean outlet ana inlet temperatures, respectively. The heat transfer area is taken as the inside surface area of the test section between the two power flanges. The heat flux was based on the heat gained by the fluid as given by equation (5). This was checked by the electrical power input corrected for the measured external heat loss. A maximum deviation between these two of 9% was observed. The heat balance for each run is shown in Table T. A sample calculation for the determination of the loeal coefficient of heat transfer is presented in Appendix I. Figure 11 shows a typical tempereture profile for the inside tube wall. The physieal properties of sodium hydroxide necessary to these calculations are presented graphically in Appendix II. The physiecal properties were evaluated at the average fluid mixed-mean temperature. Correlation of the Heat Transfer Coefficient By c¢onsidering the dimensionless differential equations describing heat transfer and fluid flow, a2 mumber of dimensionless moduli arise which characterize forced convection heat transfer in similar systems. These dimensionless parameters (Nusselt, Reynolds and Prandtl moduli) are functionally-related by the equation Nu= & {(Re, Pr) (6) UNCLASSIFIED DWG. 16331 950 900 ot ™, TUBE INNER WALL I, ‘Q-; —7 o™ =2 © g -~ | "} — / FLUID MIXED-MEAN 850}~ ( 800 0 2 4 6 8 10 12 14 16 I8 20 22 24 Distance from Entrance of Test Section, inches Fig. Il. Typical Temperature Profiles Along Axis of Electrically Heated Test Section. _ga_ 26 Reynoldslz, Prandtlll, von KérménG, Boelter, Martinelli and Jonassenl and others have determined the analytical form of the function, ¢ , for turbulent flow by comparing heat and momentum transfer within the fluid for several ideal systems approximating real systems. For convenience, equation 6 is usually written in the empirical form _ Nu = a Rel PrP (n where the constants a, n and p are experimentally determined. In general n is taken as 0.8. Dittus and Boelter> found that equation 7 could be written as Nu = 0.0243 Re0-8 PrO-* (for nheating) (8) Nu = 0.0265 Re0-8 py0-3 (for cooling) (9) McAdams? correlated data for both heating and cocling by the equation Nu = 0.023 ReC-8 pr0-4 (10) In Table I are listed the pertinent measurements and caleculated re- sults of the présent sodium hydroxide experiment. The heat transfer coefficient was ecaleculated for a position for down-stream (I/D = 100 or greater) where fully developed turbulent flow existed. Plotting the experimental data as Nu vs Pr for various values of Re, it is seen that p in equation 7 is approximately O.4k. A least squares analysis of the data then yields the result, a = 0.021. That is, the experimental turbulent forced eonvection data for molten sodium hydroxide can be correlated by the equation | Nu = 0.021 Re0:8 pyO.% (11) TABLE I EXPERIMENTAT, DATA AND CALCULATED RESULTS - DETERMINATION OF HEAT TRANSFER COEFFICIENT OF MOLTEN SODIUM HYDROXIDE ap/A w te-ty tmo-tm 1 tw h Heat Balance Run Btu/hr-£t2 Ibs/hr OF rge ot THgve Tmggve Btu/hr-£t°°F (agtq) ‘ar) Re Pr Mu 1 27,312 365 9.2 8.6 713 718 2970 0.93 6283 6.8 48.4 2 4,636 500 18.2 17.7 791 768 4100 0.97 10092 5.6 69.6 3 122,448 563 27.% 26.0 814 781 W70 0.96 11757 5.4 72.9 Iy 142,801 419 38.2 42.6 879 835 3740 0.98 10151 4.5 61.0 5 133, 8hL 410 38.1 40.2 876 830 3515 0.94 9808 4.6 57.3 6 113,604 392 30.7 36.4 895 859 3700 0.96 10019 L.2 60.3 7 122,561 ook 67.2 76.2 ol7 873 1830 0.99 5377 4.0 29.8 8 128,950 366 38.5 43.7 850 805 3350 0.94 8172 5.0 5h.6 9 124,220 ' 319 L42.5 h1.h4 861 812 2925 0.91 7258 4.9 .7 10 83, klr7 4ol 19.1 21.0 855 832 4370 0.95 11759 4.5 71.3 11 138,509 hho 32.8 38.3 879 839 4005 0.96 10957 k.4 69.0 12 134,868 336 41.3 50.7 o1k 866 3265 0.96 8758 4.1 53.2 135,258 293 6.0 59.2 o3 890 2940 0.95 8029 3.8 47.9 129,786 312 43.3 53.2 937 887 3000 0.93 8543 3.8 48.9 143,728 koo 36.3 42.5 875 832 3960 0.99 1009k 4.6 64.6 119,960 193 65.6 79.0 SLT 874 1835 0.98 5103 k4.0 30.0 UNCLASSIFIED DWG. 16332 100 2 80 60 / /, 40 £ / Nu ) Ly N pr 4 / v v ’ /‘? 20 //,/ ' %o === MCADAMS, EQUATION 4c EQUATION 7 10 2,000 10,000 60,000 Re Fig. 12. Experimental Heat Transfer Coefficients for Sodium Hydroxide. 29 for values of the Reynolds modulus between 6000 and 12000. The standard deviation of the data from this equation, assuming that the Reynolds modulus is exactly known, is 0.00l. The experimentel data are shown in Figure 12 with relation to the derived correlating equation,ll, and the McAdems equation,l0. Tt is seen that the data lie approximstely 9% below equation 10. Thermal Entrance Region It was possible to obtain from the experimental data for each runm, the thermal entrance length. This length is defined as the distance from the entrance of the test section at which the heat transfer coefficient has reached a value within a given percentage of the established value. The entrance length is normally expressed in terms of the number of tube diameters from the entrance. The local coefficient of heat transfer is shown in Figure 13 as a function of the ratio, (L/D), for a typical experimental run. Figure 14 gives the variation of the thermal entrance length, (L/D)e, with the Peclet modulus (Re-Pr). In the present investigation (L/D)e was taken as the (I/D) at which the local conductance had dropped to a value of 1.1 of the established value. Poppendieklo reviews the thermal entrance length dsta for an air system (Hunble, Lowdermilk and Desmona measurements) and for a mercury system (English and Barrett1+ measurements). Both of these experiments were conducted under conditions of uniform heat flux. DBased on the thermal entrance length criterion used in the present report, he found that for air, (L/D)e = 15 or greater, and for mercury, (L/D)e = 5 or greater. UNCLASSIFIED DWG. 16333 6,000 N . r < 5000 - \ m N - B = _4,000 \ ..u(:: Mw-—o—.wb #*fim—- © % o 3,000 \L o ? © O o | 2,000 Re= 10,000 Pr=4.2 1,000 00 20 40 60 80 i00 120 140 160 180 Pipe Diameters from Entrance, L/D Fig. 13. Local Heat Transfer Goefficients for Sodium Hydroxide Flowing in a Pipe Under Uniform Flux Conditions. UNCLASSIFIED ING. 16 50 334 40 / 30 7 20 \&*’ o (L/D)e ® 'IE— 5 10,000 100,000 Pe Fig. 14. Thermal Entrance Length for Molten Sodium Hydroxide 300,000 s 1 2 ey AT R A R 52 Analysis of Errors Error theory shows that for a function of several variables, M=f(m, mp, ..., my), the maximum error in M, A M, can be approxi- mated by the expression, Am= O¥ oM oo+ OM : m ?bnh.llflu,+ S0 Ay + + > Loy However, it is to be noted that the actual error in M might not equal the maximum error calculated, as errors in the various measured quantities might offset each other. Applying this type of analysis to the present experiment, the following errors can be tabulated: 1. For the heat flux, (qf/A), a. error in (tm,o - tm,1), 1% b. error in flow rate, w, 0.6% c. error in heat transfer area, A, 1.4% for a total error of 3% 2. For the temperature difference, (tg - tp), a. error in temperature, ty, 0.5% b. error in temperature drop through tube wall, ty - tg, 2% c. error in fluid mixed-mean temperature, Tm, for a total error of 4.5% Thus, the maximum possible error in h is 7.5%. The precision of the experiment as indicated by the standard deviation cbtained through & least squares analysis of the data is 4.8%. n,p & Q@ H P o= ~ ’ (L/D) At 33 Iv. NOMENCLATURE Constant in heat transfer correlating eguation 7 Conversion factor, 3.413 Btu/hr-watt Speeific heat of fluid at constant pressure, Btu/lb. fluid-OF Inside diameter of tube, £t Coefficient of heat transfer in region of fully developed turbulent £low, Btu/hr-ft2-CF; hx, coefficient at position x on tube. Thermsl eonductivity of fluid, Btu/hr.ft° °F/ft; kyi, thermal conductivity of nickel Constants in heat transfer correlating equation 7 Rate of heat transfer, Btu/hr; gq,, heat gained by the fluid, 9] heat loss to the enV1ronmen£ des electrical heat input Radius of tube, ft; Ty, outside wall, rg, inside wall Temperature of environment, ©F Fluid mixed-mean temperature, °F; tp i, at test section inlet; tm,0, at test section outlet; tm.ave: arithmetic average of tp i and tm,o0- Inside tube wall temperature; tg ave, average Outside tube wall temperature; ty, ave, average Mass rate of flow, 1lbs. fluid/hr Heat transfer surface, ft2 Voltage impressed on test section, volts Mass velocity, lbs. fluid /hr-(Ft° of tube cross section) Current passing through test section, amperes Distance down tube from entrance, ft. Length to diameter ratio for tube, dimensionless; (L/D)e, at end of thermal entrance region Volume of metal in tube wall, £t3 Volume heat generation, Btu/hr-ft3 Temperature difference, °F Nu Pr Re Pe 3l IV. NOMENCLATURE (CONT'D) Absolute viscosity of fluid lbs/hr.ft Density of fluid 1bs/ot> Nusselt modulus, dimensionless, (hd/k) Prandtl modulus, dimensionless, (epM /k) Reynolds modulus, dimensionless, (dG/™) Peclet modnlus, dimensionless, Re-Pr, (“Pde) 1. 10. 11. 12. 25 V. BIBLIOGRAPHY Boelter, L. M. K., Martinelli, R. C., Jonassen, F., Trans. A.S5.M.E., 63, Lu7 (1941). Corrosion Handbook, ed. Uhlig (1940). Dittus, F. W., Boelter, L. M. K., Univ. Cal. Publ. in Engr. 2, No. 13, 443 (1930). English, D., Barrett, T., Atomiec Energy Research Establishment, Ministry of Supply, Harwell, Berks, AERE E/R 547, Uneclassified, (1950). Hoffman, H. W., Doctoral Thesis, The Johns Hopkins University Library (1951) . Karman, T. von, Trans. A.S.M.E., 61, 705 (1939). Humble, L. V., Lowdermilk, W. H., Desmon, L. G., Nat. Adv. Comm. Aero., Report 1020 (1951). McAdams, W. H., "Heat Transmission"”, 2nd edition, p 168 (19L42). Poppendiek, H. F., Oak Ridge National Laboratory, Physies Report 913, Unclassified (1951). Prandtl, L., Physik. Zeitsehr., 29, 487 (1928). Reynolds, O., Proe. Manch. Lit and Phil. Soc., 14, 7 (1874); Collected Papers, 1, 81 (1900), Cembridge. 36 - APPENDIX I Sample Calculation Caleulations are based on run number 6. 1. Experimental Data (a) () (e) (d) (e) (£) (g) w = 392 lbs/hr tp,i = 840.3 OF tm,o = 876.7 °F Outside tube wall temperatures, t, E = 6.8 volts I = 324 amperes 2. Inside tube wall temperature thus, or te ¥ 100 OF _ : 2 4, = W (%2 1n z/m, - wo- Ts° tw-tg B (T Ty, 5 ) Iy = 0.01563 £t rg = 0.00979 £t ks = 30.48 Btu/br.ft.9F at ty ave ¢ EI W= =3 where ¢ = 3.4%13 Btu/hr-watt V = 0.000932 pt3 k= (3.413)(6.8) (324) (0.00048) tyts = (0.000532) (2) (30.58) == 693 OF ts = by - 6.3 A plot of tg vs. L is shown in Figure 1ll. 3. FEleetrical heat inpul ¢ EI = (3.413)(6.8)(324) 7520 Btu/hr i %e i 10. 31 APPENDIX I (CONT'D) Heat gained by fluid 4 = ¥ ¢p (B 0 - tp,1) = (392)(0.49)(36.4) 6988 Btu/hr H U Heat loss From Figure 7, q; = 239 Btu/hr at y gve - te = 795°F Heat balance , 6988 + 2 (ap + @ )/Ge = —-9-——75.30——32= 0.9 Heat flux (Qf/A)X = Tffgggggy = 113,604 Btu/hr-££° Temperature difference From Figure 11, the difference t5; - tp can be obtained for various L's. These were taken at 2,3,4,5,..., 22 inches. Heat transfer coefficient ( Qf/A) X T = )y h, was calculated for the values of (tg - tp) at the position hx = x. E.g. at L = 20 inches, tg - ty = 30.7°F and h = 3700 Btu/hr.£t> °F. h; as a function of L is shown in Figure 13. Dimensionless moduli a6 _ (0.00979)(5.25 x 106) Re = (5.139 = 10019 . _ hd _ (3700)(0.00979) _ Fo=x= (0.6) = 60.3 pr- SpH_ _ (0.49)(5.13) _ y.o k (0.0) Nu /P-4 = 3h.0 Pe = Re-Pr = 42,080 38 APPENDIX II Physical Properties of Molten Sodium Hydroxide The physical properties of sodium hydroxide were obtained from the follow- ing sources: cp : Terashkevich and Vishnerskii - J. Gen. Chem. (U.5.8.R.), 7 (1937) B : Arndt and Ploetz - Zeiltsch. fur phys. chem., 121, k39 (1926) p : Ibid k : Deem, Battelle Memorial Institute, personal commnication. For the temperature range of the experimental data reported here, & value of k = 0.6 Btu/hr-£t-°F was used. The physical properties as & function of temperature are shown in Figure 15. UNCLASSIFIED -39- DWG. 16379 0.0 q\ L 9.0 \ - 8.0 \ | . N o l o o Absolute Viscosity, g, |b./hr. ft. T o o N o 0.540 | -~ \\ \.Q\\ -4.0 N 0.520 & S & \\ \*‘ 3 < o ° (.0F3.0 0.500 _ .é \ / Efi ~ 20.9 \\ 0.480 * 5 / o . N g A N fi a 3- \ wn 0.8 0.460 o >3 - = S ; \ 0.7 5 N £ / @ 4 \\ = k \ "o.6 o~ o -\ 600 700 800 900 1,000 1,100 Fig. I5. Variation of Temperature, °F Physical Properties of Sodium Hydroxide with Temperature Lo APPENDIX III Derivation of Equation for Temperature Drop Across a Tube Wall in which Heat is being Generated The trensient conduction equation for the case in which heat generation exists within the solid is 2t _ 2 W S5 =avV t+ e (1) where @ = thermal diffusity = 53__ vz'b = Laplacian of ¢ W = source term in Btu/hr.ft3 Then for steedy state conditions, _-g-g = 0, and uniform heat generation, W = constant. For a circular tube, 'the Laplacian as expressed in e¢ylindrical coordinates is vat = .._.a_z..t;_ + .];a_.J.G..l.__a.._e_-E 4 !'.....aet 3 TOT P 1'23@_2 Since the temperature distribution is symmetrical with re.épect to @,%2; =0 and since beyond the entrance region, -aa-g = const, equation 1 thus reads 0k [a% 1 _at W e \a® T dr) pop or Bt 1 at W g T r ok - (2) if let s = _8% dr Then, equation 2 becomes ds 1 W _ T=FrTE®T " % (3) APPENDIX ITI(CONT'D) This is of the form fi& 3 i Hi+ where P = fl(x) il 1 = S~ 0y Q= fe(x) which has the solution, then dex:fraJ—'dI‘:'lnr e-JPAx _ o-Imx _ 1 r o-JPax _ lor _ thus, _ 1 LW S-E(Irfidr+cl) =..J.'. ‘-E _I.:.?:__'FC) 7 k 2 or @ _ _¥r C dr- k2" r The heat flow through the surface defined by Tr = rw is dt (@/B)y = - (dr) r=r, or (_(}E) (9/A)y dr P—rw = - ' then, substituting this boundary condition into equation 10, PR A ——r—— + e X x 27 ¥, 41 (4) (5) (6) (7) (8) (9) (10) (11) (12) L2 APPENDIX III(CONT'D) Substituting for C in equation 10, dt = _Wry (r2W _r (a/h)y]| 1 dr k 2 2k k T - -W_ [r. rwz‘i - (a/A)y r.. 1 (13) r k W r integrating equation 13 5 rw rw ty = tg=-W_ [ 22 - {a/A)y r In r ot X (2 W in r) re k¥ re rwz -r.2 T Ty - (-——-—-—-—-—-*‘-’-—-r.,,zlnr—:) - (a/adw T g (L) k ok 2 For the particular problem under consideration %% at r = r, is essentially zero, Thus, equation 14 reduces to W r, . r.2-r.2 by - by = o <;w2 1n ?g W 5 s /) (15)