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A laboratory study of thermal convection under a central force field Chandra, Bhuvanesh 1971

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IIZII  A LABORATORY STUDY OF THERMAL CONVECTION UNDER A CENTRAL FORCE FIELD BY  BHUVANESH CHANDRA  B.Sc,  U n i v e r s i t y o f Lucknow, 1961  M.Sc, M.Sc,  I n d i a n I n s t i t u t e o f Technology, K h a r a g p u r , 1965 U n i v e r s i t y o f Western O n t a r i o , 1969  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of GEOPHYSICS We a c c e p t t h i s t h e s i s a s . c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA November, 19 71  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may by h i s r e p r e s e n t a t i v e s .  be  granted by  permission.  Department of  Geophysics  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  December 27,  1971.  Department or  I t i s understood t h a t copying or  of t h i s t h e s i s f o r f i n a n c i a l g a i n written  the Head of my  Columbia  s h a l l not be  publication  allowed without  my  (i) ABSTRACT  T h i s t h e s i s p r e s e n t s the r e s u l t s o f a t h e o r e t i c a l  and  e x p e r i m e n t a l s t u d y of t h e r m a l c o n v e c t i o n under the i n f l u e n c e o f a central force f i e l d .  Flows i n the atmosphere and i n the  of the e a r t h are thought Coriolis  to o c c u r under a n e a r b a l a n c e between  and buoyancy f o r c e s .  Thus, a d e s i r a b l e model o f  f l o w s would i n c l u d e s p h e r i c a l symmetry i n the f o r c e f i e l d rotation.  core  these and  The p r e s e n t s t u d y , i n w h i c h c o n v e c t i o n under a c e n t r a l  f o r c e f i e l d i n c y l i n d r i c a l geometry has been a c h i e v e d , i s the f i r s t s t e p towards such a model. The  system c o n s i s t s o f a c o o l o u t e r c y l i n d e r and a h o t  inner cylinder with a d i e l e c t r i c l i q u i d ( s i l i c o n e o i l ) f i l l i n g the annulus between them. vertical.  The  The  common a x i s o f the c y l i n d e r s i s  i n n e r c y l i n d e r i s grounded and the o u t e r  i s k e p t a t a h i g h a l t e r n a t i n g (60 Hz) p o t e n t i a l .  cylinder  This intense  a l t e r n a t i n g e l e c t r i c f i e l d p r o v i d e s the r a d i a l buoyancy f o r c e which r e s u l t s i n c o n v e c t i v a heat t r a n s f e r at a c e r t a i n temperature The  critical  gradient.  f l u i d i n the system i s found t o behave l i k e  f l u i d i n a g r a v i t a t i o n a l f i e l d , heated  from below.  a l a y e r of  Below a  c e r t a i n c r i t i c a l v a l u e o f a d i m e n s i o n l e s s number ( e q u i v a l e n t to  (ii) the Rayleigh number with the e l e c t r i c a l force substituted f o r gravity) there i s no convective heat transfer.  Above the c r i t i c a l  value, flow sets i n with the convective heat transfer proportional to the modified Rayleigh number.  Marginal s t a b i l i t y analysis  gives a c r i t i c a l e l e c t r i c a l Rayleigh number i n agreement with the experimentally determined value.  (iii) TABLE OF CONTENTS ABSTRACT  (i)  LIST OF FIGURES  (v)  LIST OF TABLES  (vii)  ACKNOWLEDGEMENTS CHAPTER I  (viii)  GENERAL INTRODUCTION  1.1  Introduction  1.2  Thermal convection and geophysical  1.3  .1  applications  4  Scope of this thesis.  7  CHAPTER I I THEORY 2.1  Introduction  9  2.2  The perturbation, equations  11  2.3 -The p r i n c i p l e of exchange of s t a b i l i t i e s 2.4  The solution f o r the case of a narrow gap  2.5  23  27  The solution for a wide gap when the marginal s t a t e . i s stationary  .  28  CHAPTER.Ill EXPERIMENTAL ARRANGEMENT AND RESULTS 3.1  Introduction  37  3.2  Experimental arrangement  40  3.3. .The experimental  procedure  48  (iv) 3.4  Errors and corrections  54  3.5  The experimental results  63  3.6  Accuracy  65  CHAPTER IV  DISCUSSION AND CONCLUSIONS  4.1 Discussion of results  66  4.2  67  Conclusions  REFERENCES  68  APPENDIX I TABLES OF THERMISTOR CALIBRATION  71  APPENDIX.II TABLES OF HEAT TRANSFER, MEASUREMENTS  74  APPENDIX I I I TABLES OF CORRECTED DATA  83  (v) LIST OF FIGURES  FIGURE 1  Coordinate system and c y l i n d r i c a l cavity.  10  FIGURE 2  Dependence of the numerically determined e l e c t r i c a l Rayleigh number on the dimensionless wave number.  32  The v e l o c i t y p r o f i l e .  33  FIGURE 4  The temperature p r o f i l e .  34  FIGURE 5  The stream l i n e s at the onset of i n s t a b i l i t y .  35  FIGURE 6  A cross section of the apparatus.  39  FIGURE 7 .  The c i r c u i t f o r the heating c o i l .  41  FIGURE 8.,  C i r c u i t . f o r the bridge.  44  FIGURE 9  Kinematic v i s c o s i t y as a function of temperature.  45  ... D i e l e c t r i c constant as. a function of temperature.  46  FIGURE 3  FIGURE 10 FIGURE 11  FIGURE 12 FIGURE 13 FIGURE 14 FIGURE 15 FIGURE 16  .  Dependence of the temperature of the outer cylinder less mean of i n l e t - o u t l e t temperatures on heat transfer rate.  49  Calibration of Ammeter 1^ against a laboratory standard.  51  Calibration of ammeter I standard.  52  against  Calibration of voltmeter against standard.  a laboratory a laboratory 53  Dependence of temperature difference on heat ...... transfer rate f o r 0 kv rms between the cylinders.  55  Dependence of temperature difference on heat transfer rate f o r 4.06 kv rms between the cylinders.  56  (vi) FIGURE 17  Dependence of temperature difference °n heat transfer rate f o r 6.00 kv rms between the cylinders.  FIGURE 18  Dependence of temperature difference, on heat transfer rate f o r 6.92 kv rms between tjhe cylinders.  FIGURE 19  Dependence of temperature difference heat transfer rate f o r 7.91 kv rms between fche cylinders.  FIGURE 20  Dependence of temperature difference on heat transfer rate f o r 9.63 kv rms between the cylinders.  FIGURE 21  Dependence of temperature difference on hpat transfer rate f o r 10.15 kv rms between the cylinders.  FIGURE  22  Dependence of temperature differej^pe on heat transfer rate f o r 10.80 kv rms between t^he cylinders.  FIGURE  23  Nusselt number as a function of e l e c t r i c a l Rayleigh number.  (vii)  LIST OF TABLES Table 1  E l e c t r i c a l body force at the inner and outer cyliners for given voltages across the gap.  22  Table 2  Dimensions of the c e l l .  42  Table 3  Thermistor calibrations f o r the inner cylinder.  72  Table 4  Thermistor calibrations f o r the i n l e t and outlet points.  73  Table 5  Typical properties.of s i l i c o n e o i l .  48  Table 6  Observed data at 0 kv rms.  75  Table 7  . Observed data at 4.06 kv rms.  76  Table 8  Observed data at.6.00 kv rms.  77  Table 9  Observed data at 6.92 kv rms.  78  Table 10  . Observed data at 7.91 kv rms.  79  Table 11  Observed data.at 9.63.kv rms.  80  Table 12  Observed data at 10.15 kv rms.  81  Table 13. , Observed data at 10.80 kv rms.  82  Table 14.  Corrected data f o r 0 kv rms.  84  Table 15  Corrected data f o r 4.06 kv rms.  85  Table 16  Corrected data f o r 6.00 kv rms.  86  Table,17,-  Corrected data f o r 6.92 kv rms.  87  Table 18  Corrected data.for 7.91 kv rms.  88  Table 19  Corrected data for 9-63 kv rms.  89  Table 20  Corrected data for 10.15 kv rms.  90  Table 21 .  Corrected data for 10.80 kv rms.  91  (viii)  ACKNOWLEDGEMENTS  The author i s indebted to Dr. D. E. Smylie for d i r e c t i n g the research reported i n this thesis. the thesis c r i t i c a l l y :  The following persons read  Dr. D. E. Smylie, Dr. G. K. C. Clarke and  DR. T. J . Ulrych, a l l of the Geophysics Department, and Dr. P. H. LeBlond of the I n s t i t u t e of Oceanography.  Mr. H. Lau of the Mechanical  Engineering Department contributed to many discussions. F i n a l l y , the author wishes to acknowledge the assistance of the s t a f f members and the graduate students of the department who contributed i n various capacities. The work was financed by grants from the National Research Council of Canada to Dr. D. E. Smylie.  CHAPTER I GENERAL INTRODUCTION  1.1  Introduction In the l a s t few decades, e f f o r t s have been made to study  motions i n the Earth's atmosphere, oceans, mantle and core the use of laboratory models (Fultz, 1961; Long, 1954).  through  The incentive  for the use of .models comes from the d i f f i c u l t i e s of solving the equations of motion for general convective flow problems.  At the  present time, the equations governing the general flows can only be solved approximately under i d e a l i z e d conditions.  Thus, laboratory  models have become, important i n geophysical f l u i d dynamics. The large scale c i r c u l a t i o n of the atmosphere occurs under the combined  influence of a spherically symmetric g r a v i t a t i o n a l f i e l d  and the rotation ..of the Earth.  .It i s suspected-that a. roughly s i m i l a r  s i t u a t i o n prevails. i n the Earth!s l i q u i d core.... Thus, a laboratory model of these flows.should .include both.the s p h e r i c a l symmetry of the g r a v i t a t i o n a l f i e l d and the rotation of the Earth.  The e f f e c t of  .rotation has been-the subj.ect..of study, of many, authors (for recent references, see. Greenspan,. 1968). . In these ..studies , certain flow regimes have been recognized which seem to have their counterpart i n the atmosphere i n the form of large scale eddies or cyclones. However, no .laboratory model which simulates a s p h e r i c a l l y symmetric  2 g r a v i t a t i o n a l f i e l d has y e t been developedo  I n the absence o f a  s u i t a b l e method o f s i m u l a t i n g a g r a v i t a t i n g  f l u i d sphere i n the  l a b o r a t o r y , t h e v a l i d i t y o f the r o t a t i n g models i s l i m i t e d t o t h e . p o l a r r e g i o n s where s t r o n g C o r i o l i s f o r c e s a r e p r e s e n t as i n the core and the atmosphere„  I n t h e t h i n s h e l l o f the atmosphere,  however, t h i s d e f e c t i s n o t so s e r i o u s because the f l o w i s e s s e n t i a l l y two d i m e n s i o n a l  as opposed t o the core f l o w which i s t h r e e ,dimensional.  T h i s t h e s i s i s the outcome o f a p r e l i m i n a r y  investigation  f o r s e t t i n g up a l a b o r a t o r y model o f g r a v i t a t i o n a l c o n v e c t i o n -  Smylie(1966)  f i r s t s u g g e s t e d the use o f s t r o n g e l e c t r i c f i e l d g r a d i e n t s i n d i e l e c t r i c l i q u i d s t o s i m u l a t e the c e n t r a l n a t u r e o f the g r a v i t a t i o n a l field.  The e l e c t r i c a l p r o p e r t i e s o f a . l i q u i d a r e f u n c t i o n s o f  temperature„  For a nonpolar  l i q u i d the v a r i a t i o n o f d i e l e c t r i c ,  c o n s t a n t w i t h temperature i s due s o l e l y t o t h e change i n d e n s i t y (Debye, 1929, p 27)„  Thus, i n the p r e s e n c e o f temperature  i n the d i e l e c t r i c l i q u i d , . a n buoyancy force„ i n the f l u i d o  intense e l e c t r i c f i e l d results i n a  T h i s buoyancy f o r c e w i l l produce an i n s t a b i l i t y However,.the v i s c o s i t y , t h e r m a l , c o n d u c t i v i t y and the  b o u n d a r i e s w i l l a c t t o s t a b i l i z e the f l u i d and s i g n i f i c a n t heat t r a n s f e r  variations,  convective  s h o u l d o c c u r o n l y when the temperature g r a d i e n t i s  appreciable,, I n d i e l e c t r i c l i q u i d s o f moderate d i e l e c t r i c c o n s t a n t , the t h e r m a l c o n v e c t i o n produced by the e l e c t r i c buoyancy f o r c e i s analogous to g r a v i t a t i o n a l c o n v e c t i o n , where the i n s t a b i l i t y o c c u r s as a r e s u l t o f h e a t i n g from below ( o r c o o l i n g  from above).  I n c o n t r a s t t o the u b i q u i t o u s  3 and unalterable g r a v i t a t i o n a l f i e l d , however, e l e c t r i c f i e l d s can e a s i l y be generated and shaped i n the laboratory by e x i s t i n g techniqueso In a steady e l e c t r i c f i e l d space charges accumulate even i n a good d i e l e c t r i c and the c i r c u l a t i o n of the l i q u i d i s due to the movement of these charges.  The free charge buildup occurs  exponentially i n time with a time constant e/a where e i s the p e r m i t t i v i t y and a i s the conductivity of the f l u i d . i s known as the e l e c t r i c a l relaxation time.  This constant  I f an alternating  f i e l d i s applied at a frequency much higher than the reciprocal of  the relaxation time, free charge does.not have time to accumulate.  Thus, i n an alternating f i e l d , the d i e l e c t r i c effects dominate the free charge e f f e c t s o  The e l e c t r i c body force depends on the  gradient of the square of the e l e c t r i c f i e l d *  For most.dielectric  l i q u i d s (for example, transformer o i l and s i l i c o n e o i l ) , 60 Hz i s a s u f f i c i e n t l y high frequency to prevent the buildup of free charge. At  the same time, d i e l e c t r i c loss at 60 Hz i s so low that i t makes  no s i g n i f i c a n t contribution to the temperature f i e l d .  Further,  variations i n the body force are so rapid that i t s mean value can be assumed i n determining f l u i d motions, except i n the case of l i q u i d s of extremely.low v i s c o s i t y . . Both theoretical and experimental study of electroconvection (convection under steady f i e l d s ) has been reported by many authors (for example, Kronig and Schwarz, 1949; Turnbull, 1968; Malkus and Veronis,.1961; Melcher and Taylor, 1969; etc.).• The electroconvective  4 i n s t a b i l i t y occurs even i n the absence of temperature v a r i a t i o n s ; Avsec and Luntz (1937) have observed two dimensional t o r o i d a l motions i n a d i e l e c t r i c l i q u i d f i l l i n g the gap between two concentric cylinders.  When the inner cylinder i s earthed and the e l e c t r i c a l  p o t e n t i a l of the outer cylinder i s raised, then at some c r i t i c a l p o t e n t i a l difference, steady c e l l u l a r patterns are formed i n the fluid. Recently, Gross(1967) and Gross and Porter(1966) have suggested the.use of space charge effects i n modelling geophysical flows.  In these models, charge transport becomes the analogue of  heat transfer and. i f .temperature gradients are present, heat w i l l be advected by the .charge induced flow.  The mechanisms of charge  generation and transfer at the electrodes, and the nature of the process by which charge motipn s t i r s the f l u i d are not w e l l understood.  Therefore, the q u a l i t y of analogy to geophysical phenomena  i s not clear. 1.2  Thermal convection and Geophysical applications; Although the phenomena of thermal convection as a mode of  heat transfer was discovered i n the 18th century, the f i r s t quantitative experiments on convection were done by Be"nard(1900). found that  He  i f a thin layer of l i q u i d , free at i t s upper surface, i s  heated uniformly at the lower surface, a regime of polygonal convection c e l l s i s formed as soon as a certain c r i t i c a l temperature gradient i s reached...The walls of these c e l l s are v e r t i c a l and the movement of the l i q u i d i s upward i n the centre  and downward at the periphery.  5 Rayleigh(1916) l a i d the t h e o r e t i c a l foundation for the explanation of Benard's r e s u l t s .  I t appears now, however, that the  convection i n Benard's experiments was actually p a r t i a l l y driven by the variations i n the free surface conditions and not e n t i r e l y by the buoyancy forces as Rayleigh assumed.  In l a t e r experiments  (see for example Chandra, .1938; .Schmitd and Milverton, 1935; and Silveston, 1958), the free surface has been eliminated by placing an isothermal l i d on top of the f l u i d layer.  These experiments also  showed the development of regular hexagonal c e l l patterns.  The  c r i t i c a l temperature gradient at which .convection c e l l s develop i increases rapidly as the layer becomes thinner.. Several modifications of Rayleigh's  theory  (for example, J e f f r e y s , 1926; Pellew and  Southwell, 1940; Low, 1929; etc.) to f i t this case give excellent agreement with the experimental  observations.  Rayleigh's approach  to the problem ( i n f i n i t e s i m a l , perturbations on an equilibrium state) s t i l l remains the basis of the modern treatment of the marginal s t a b i l i t y problems (Chandrasekhar, 1961, chapter 2). It i s believed that many flows of geophysical i n t e r e s t are driven by thermal convection.  I t has been suggested that large  scale convection currents e x i s t i n the garth's mantle i n attempts by many authors  to i n t e r p r e t certain topographic features of the  Earth's surface (Vening-Meinesz, 1962).  Recently, McFadden(1969)  has considered the e f f e c t of a region of low v i s c o s i t y on thermal convection i n the Earth's mantle to try to explain how convection  6  i n only the upper mantle could lead to flows of continental h o r i z o n t a l scale. The presence of thermal convection i n the atmosphere was suggested by Hadley (1735); d i f f e r e n t i a l heating between d i f f e r e n t latitudes gives r i s e to ascent of a i r i n t r o p i c a l regions.  This leads  to a flow of a i r towards the equator at lower levels and a flow away from the equator at higher l e v e l s .  The combined effects of  surface f r i c t i o n and the rotation of the earth deflects these currents and thus, according to Hadley's o r i g i n a l theory, easterly winds form at lower levels and westerlies at higher l e v e l s .  From  this idea, the w e l l known c e l l u l a r model f o r the c i r c u l a t i o n of the whole atmosphere has been derived (Rossby, 1941). The geomagnetic f i e l d i s now considered to originate from e l e c t r i c currents produced by inductive i n t e r a c t i o n between hydrodynamical motions i n the Earth's conductive f l u i d core and a small adventitious magnetic f i e l d .  Thus, a s a t i s f a c t o r y theory f o r the Earth's magnetism  requires a s a t i s f a c t o r y theory f o r the hydrodynamics of the core. Several speculations have been made regarding the energy source that drives the f l u i d motions i n the core.  Among them thermal convection  i s one of the p o s s i b i l i t i e s (Bullard and Gellman, 1954). Oceanographers have long studied the c i r c u l a t i o n of the oceanic currents.  The ocean currents are believed to be the r e s u l t  of the combined e f f e c t s of the thermohaline motions (density inhomogeneity caused by temperature and/or s a l i n i t y differences) and the wind driven motions.  The former are thought to be more important  7 i n deep water.  However, unlike the case of the atmosphere, the  thermal convection  i n the oceans occurs, not because of heating from  below, but because of cooling from the upper surface. circulation  The  thermohaline  originates as a v e r t i c a l flow sinking to mid-depth or  even to the ocean bottom, followed by horizontal motion. 1.3  Scope of the present work This thesis i s intended to demonstrate the f e a s i b i l i t y of  using an intense alternating e l e c t r i c f i e l d acting on a d i e l e c t r i c l i q u i d to produce a s p h e r i c a l l y symmetric force f i e l d .  A buoyancy  force i s produced by the v a r i a t i o n of d i e l e c t r i c constant with temperature.  Since i t i s a f e a s i b i l i t y study, the experiment  set up i n cylindrical.geometry  to.simplify .construction.  was  However,  the techniques, learned here can be carried d i r e c t l y over to the spherical-case, which i s the r e a l geometry for large scale  geophysical  flows. . . The system consists of two v e r t i c a l concentric cylinders of radii  r^ and  r^ ( r^ j_) • >r  The .space between the cylinders i s f i l l e d  with a d i e l e c t r i c l i q u i d ( s i l i c o n e o i l ) .  The inner cylinder i s  maintained at a fixed temperature T^ which i s higher than the constant temperature T2 of the outer cylinder.  In addition to a temperature  gradient, a radial., alternating e l e c t r i c f i e l d i s also imposed be tween . the c y l i n de rs. In chapter I I , the marginal s t a b i l i t y equations for this  8 problem are derived under the assumptions approximation. for this theory.  The effect  of the Boussinesq  of the Earth's gravity has been ignored  The solution of the marginal s t a b i l i t y equations  has been found numerically and i t i s shown that the convection induced by the e l e c t r i c a l body.force occurs only when the dimensionless number (equivalent to the Rayleigh number with .the e l e c t r i c buoyancy force substituted f o r gravity), exceeds a.certain c r i t i c a l value. When the gap thickness i s very much smaller than the mean radius of the cylinders ( i n this l i m i t i n g  case boundaries take the form of  plane surfaces, rather than c y l i n d r i c a l surfaces),,the marginal s t a b i l i t y equations are shown to reduce to the case of the Be"nardRayleigh i n s t a b i l i t y . In.chapter I I I , the experimental arrangement, technique and results of e l e c t r i c a l l y induced convection i n a v e r t i c a l annulus are described.  The onset of thermal convection was detected  by heat transfer and temperature measurements.  The f i n a l experimental  results, are shown,in the form of a graph of the Nusselt number against the e l e c t r i c Rayleigh number. The.study i s summarized i n the f i n a l chapter.  9 CHAPTER II THEORY  2.1  INTRODUCTION The problem to be studied i s the thermal convection occuring  under the e l e c t r i c a l body force i n the enclosed annular region formed by two v e r t i c a l concentric cylinders of r a d i i . r and 1  This geometry i s shown i n figure 1.  (r^r^).  Conventional c y l i n d r i c a l coordinates  (r,6,z) are indicated with z v e r t i c a l l y upwards. The inner wall of the c y l i n d r i c a l cavity i s held at a fixed temperature T.^ which i s . greater than the constant temperature T^ of the outer w a l l .  In.addition to. a temperature difference, there i s also an  alternating difference of e l e c t r i c a l p o t e n t i a l between the cylinders. In the absence.of any temperature v a r i a t i o n s , an e l e c t r i c body force is.produced in.the liquid,.and i n an equilibrium s i t u a t i o n , this body force i s entirely, balanced by the generation of a pressure gradient (Landau and Lifshitz,, .1960, pp 64-69).  The presence of temperature  variations i n the l i q u i d results i n buoyancy forces whereby warmer l i q u i d has,a tendency to- seek regions of less intense e l e c t r i c f i e l d , cooler, l i q u i d has a.tendency to.seek regions.of more intense e l e c t r i c field.  However,-this natural tendency on the part of the f l u i d w i l l  be i n h i b i t e d by i t s own v i s c o s i t y .  Thus, i f the temperature difference  FIGURE  1.  CO-ORDINATE  SYSTEM  AND  THE  CAVITY.  11 between the v e r t i c a l bounding walls i s s u f f i c i e n t l y small, the heat supplied at the inner w a l l i s transferred to the outer wall by conduction alone.  For greater temperature differences, the e l e c t r i c buoyancy  force i s s u f f i c i e n t to overcome viscous d i s s i p a t i o n and the f l u i d assumes a convective motion.  In other words, we expect that the  temperature difference between the cylinders must exceed a c e r t a i n minimum value before convective heat transfer i s r e a l i z e d . The problem under consideration, then, i s to solve the marginal s t a b i l i t y equations  for an incompressible  f l u i d layer bounded  by v e r t i c a l c y l i n d r i c a l walls at d i f f e r e n t temperatures, i n the presence of a r a d i a l e l e c t r i c a l buoyancy force. 2.2  The Perturbation  Equations  The following simplifying assumptions are made i n deriving the perturbation equations: (1)  The f l u i d i s incompressible and the density constant, except as i t modifies the electric-buoyancy force term.  (2)  The mean square e l e c t r i c f i e l d and the temperature d i s t r i b u t i o n i n the annulus are functions of radius only.  (3)  The cylinders are of i n f i n i t e extent, so that the end effects can be ignored.  (4)  The motion i s slow and the components of v e l o c i t y are small enough so that i n a f i r s t approximation, be neglected.  t h e i r products and squares can  12 The kinematic v i s c o s i t y and the ,.dif f u s i v i t y of the f l u i d are true constants. The various perturbations are axi^symmetric and thus independent 0.  of  In deriving the. marginal s t a b i l i t y .equations, the e f f e c t of the earth's g r a v i t a t i o n a l f i e l d can be ignored.  S t r i c t l y speaking i t  i s .not so.. Batchelor. (19.54). showed, that-in .a.narrow cavity between v e r t i c a l , boundaries a t d i f f e r e n t ..temperatures, there i s a- slow v e r t i c a l flow but the heat transfer across the cavity i s due mostly to conduction.  Thus, the present problem i s unstable  for a l l , temperature gradients since the earth's gravity is. always present..  The. maximum Grashof number involved .in the experiment  (described, in. .the next chapter* was about 7.5.  This i s roughly  3 .10, .times, l e s s than the ..Grashof. number....at which free convection sets -in.. (Vest, and Arpaci, . 1969)...... The gravity induced, flow i s l i k e l y to be very small., at. the.. Grashof ..number involved i n the experiment.  I t would, be an important generalization .to carry  through a solution taking into account the i n t e r a c t i o n of the convection, induced by the e l e c t r i c a l forces with the v e r t i c a l g r a v i t a t i o n a l base flow,,, but this d i f f i c u l t problem has not been tackled here. Consider that i n i t i a l l y there are no motions.  Thus, the i n i t i a l  13 state i s characterized by U = 0 T = T(r)  (1)  where U i s the v e l o c i t y vector and I i s the temperature. In'the absence of any .motions, the hydrostatic equation -grad p  o  + 1  o  = 0  reads (2)  -»•  where f -is . the e l e c t r i c body, force per unit volume at density p  o  and temperature T : p i s the pressure o o  ,  . The .temperature distribution...is governed by the equation  r  distribution,  .  <» The s o l u t i o n of equation  (3) appropriate to the boundary conditions  gives T dT dr  T  T  1  Q  r  „  2  1 r  in/i.  (4)  "1 The  solution f o r the e l e c t r i c f i e l d d i s t r i b u t i o n E i s o E j = (E ,0,0),  (5)  q  . o  r  r 2  (6)  where V i s the root mean square value of the applied p o t e n t i a l q  difference.  14 Let  the perturbed state be characterized by the v e l o c i t y  components  V  V  u , e  body force f =  + t\  (8)  Q  + p ,  (9)  E  = E  p = p  density  electric field  (7)  + E ,  (10)  o T = T  temperature and pressure  Q  p = P  Q  i + T ,  (11)  i + P ,  primes i n d i c a t i n g flow induced quantities. The. general equation f o r the body force resulting from e l e c t r i c f i e l d s i n f l u i d d i e l e c t r i c s .is developed by Landau and L i f s h i t z (1960,,p.68) -  They showed that i n an uncharged f l u i d d i e l e c t r i c of  uniform .composition,.,the . e l e c t r i c body force, per unit volume i s given by V=  1 / 2 p  o  &  r a d [ E  o (-If  }  T  "  ]  1/2E  d  If  (  )p ^ a d T  o  o  ,  (12)  o  where e i s the e l e c t r i c p e r m i t t i v i t y of the f l u i d . In the perturbed state at temperature T and density body force.per unit 1  is  Aolume  = 1/2 pgrad  [E  p , the  2  ( -g- \^  - 1/2 E  2  ( | | ) grad T p  (13)  15 The p e r m i t t i v i t y of a substance i s a function of density and temperature.  If the perturbations are small, the p e r m i t t i v i t y at  temperature T and density p can be expressed i n terms of p e r m i t t i v i t y at the reference temperature T expansion.  q  and density p  Q  by w r i t i n g a Taylor  Thus,  $ v'  +1/21  \-*.  2 + 2 c  o  3p  +c  p,r  d  \  o o  3i  + higher order terms Therefore, we can write ,  3e'  3£ .  (  ^ 3p T ;  S  .  3p T  o  '  2  ,  2  .  3 e  o  .  2  ^ 8p8T  P  ;  p  ]  o  "  dc .  - 2 T 3p .  ;  1  T  (14)  , ,T  00  and  3 1  p  3  1  p  o  3T  2  P  o  3p3T  p  o' o T  Combining (8), (12), (13) and (15), we get to f i r s t order i n the flow-induced q u a n t i t i e s ,  f*  - l/2p'grad[E  2 o  (  )  ] - 1/2E  Q  (|| )  2  o + p grad[(2 .^) o  o  ( f£- )  gradT' o  ] - ( ;.I')  T  E  (|| >  o .  +  l/2p grad[E o  - 1 / 2 E / I  2 o  (4 3T  {(^f 3p  >  p  o  )  p'  T  g  radT  o  o +  (  )  o  *' + ( f e )  T'} ] o  0  o  o  o  »']«»«.  <»)  16 Chandra (1969) showed that for a f l u i d d i e l e c t r i c of moderate d i e l e c t r i c constant, the flow-induced changes i n the e l e c t r i c f i e l d can be neglected. temperature  The order of magnitude of the second density and  c o e f f i c i e n t s of p e r m i t t i v i t y for f l u i d s i s very small  and can also be taken to be zero. . T h u s t h e . e l e c t r i c buoyancy force per unit volume can be written as . with a  f  - 1/2 p' ( f ^ - >  T  gradE  2 Q  --i°-  ( || )  p  grad' T'  (17)  p' - - ctpT'  (18)  5  denoting the c o e f f i c i e n t of thermal expansion. The equation (17) i s quite general and can be applied to a  f l u i d d i e l e c t r i c .of any shape.. . Using equation .(6),  the e l e c t r i c  buoyancy force per unit mass f o r the c y l i n d r i c a l case under consideration can be written as v  2 r [ln-^] 2  r  Sp 'T  l  J r  »  r (19)  where r i s the unit radius vector. Having found.an expression f o r the e l e c t r i c buoyancy force per. unit mass, the l i n e a r i z e d perturbation.equations:can be written as follows: (i)  momentum  17  at""  e "-I 1  v [ v  <  21)  u  r 2 Ci  TT z  V  -  / £ l x 4 . o2 °  9  i /.  TT  o (In  1 , 3e . — r  (ii)  3T*  l  continuity  3U  U  3U  dr  r  dz  (iii)  1 r  r  energy 2 r ln — l  T  3t  r  \ r  T  _  KV*T'  (24)  r  2 where V has the meaning 2 2 V = - — + - — + - — 2 r 3r 2 3r 3z 2  +  a  +  K  (25)  J  a  and v and K are the kinematic v i s c o s i t y and the thermal d i f f u s i v i t y of the f l u i d . By analysing the disturbance into normal modes, the solutions of equations (20)-(24) are to be found i n the form U  = e U(r) pt  r  U  coskz,  = e V ( r ) coskz, p t  T  = e 0 ( r ) coskz, p t  p* = e f i ( r ) coskz, p t  P  U  1  V(26)  o  = e W(r) coskz. pt  z  where k i s the wave number of the disturbance i n the a x i a l d i r e c t i o n  18 and p Is a constant which may form (26), equations  be complex.  For solutions of the  (20)-(25) reduce to  V  2  v(DD,-k - /v)U + — 2 - [a( f (In- -) l 2  P  2  - -f-  ), %  ( j§  )  \  jf  ] - DO,  (27)  2  r  (DD  A  - k  2  - p/v)V = 0,  (28) V  v(D*D-k -p/v)W + -fr  2  °— r (In — ) l  2  2p  ( || ) 3T  p  *f 2  = -kfl,  (29)  r  r  D^U  = -kW,  (30)  T -T 2 , „ - - -1^ -2 f (D^D-k"-p/K)0 v  i cil n r  and  V  U^ ,  (31)  2  — l  = D^D-k ,  2  (32)  2  where  D = —dr  * - h  a n d  D  (33)  r'  +  ( 3 4 )  For the case when p=0  the solution of equation (28) i s  V = A I ( k r ) + BK (kr) Q  where I  o  and K  (35)  Q  o  are modified Bessel functions of zeroth order,  Substituting the boundary conditions that on r i g i d boundaries  V=0,  we get A = B = 0. Hence,  V = 0.  This shows that f o r the form of assumed solutions, there cannot  (36)  19 9,  be any v e l o c i t y perturbation along  Thus, the basic flow i s i n  the form of ring v o r t i c e s . Eliminating W between equations (29) and (30), we get - | (D I)-k -p/v)D,,U  - ^ T  2  s  +  C|f)  p  if.-kfi  (37)  Substituting the above expression f o r o, i n equation (27), we get after some.simplifying steps V (DD,-k -p/v)(DD,- )U- - 2 -  2  2  2  2  k  Io(  .|£  > - ± ( || > ] % T  (ln -*x2  r  For f l u i d s of uniform composition, the p e r m i t t i v i t y may be regarded as a function of ..the state variables alone and we may write  M'.-'I't-M'p  (39)  Thus, equation -(38). reduces to 2 -~ \ £ < ||) ^ (In/) 1 Equation (40) must be solved together with (DD^-p/vXDD.-k )^ 2  (40)  2  T -T  (D.D-k -p/<)0 = - —±—r 2  r  <ln r  2 — l  '7 •  (41)  Equations (40) and (41) are the marginal s t a b i l i t y equations, the solution of which must be found under the boundary conditions • U =..DU = 9 » 0 at p r ^ and r * ^ . 8  (42)  20  Dimensionless  parameters to describe solutions i n general  form may be obtained by non-dimensionalizing suitable manner.  the equations,in a  A convenient set of non-dimensional variables i s  *\  r - r.  J2£  >  (43)  a =» kd A »  £  where the cavity gap  -  i s written as d.  through (42) become  22 a d  (DD -a -o)(DD -a )U 2  2  #  f2  #  I l n  T  (D.D-a -Pc)G -•-  r  The equations  T  2  d  r  2  r  ] 2  r  __1 3  x  2  x  (1+X?)  '•  ^  ,  3e > 9 T  e P (1+AO  (40)  (44) 3  (45)  •cin U = DU •» 9  D and  0  at C  B  0 and 1.  (46)  now have the following meaning  de • :  (47)  (48) and P (» — ) I s the Prandtl number*  It i s convenient  to make the transformation  KT (Tj-T^d  r I nl —  2  9  r  +6,  where 0 now has the dimensions of v e l o c i t y . The marginal s t a b i l i t y equations become (DD.-a -o)(DD.-a )U= -Ra 2  2  2  9  (1+U) and  (D*D-a -Pa)0 =  ^  2  R may be interpreted as the e l e c t r i c a l Rayleigh number ag Bd R =  4  VK  y 2  where  1  o  *e " "  3 2,2 ^ ( I n — ) r  p  a  , je_ . 8  T  P  n  can be regarded.as an e l e c t r i c a l l y derived .gravity at the surface the inner cylinder and  T -T 1  6.r  l  l  n  r  2 2  ^  i s the temperature gradient at the inner cylinder surface. C l e a r l y , the e l e c t r i c a l gravity, g^,  i s a strong function  o f radius and varies inversely as the cube o f . r a d i u s .  22 Table 1 The e l e c t r i c a l body force at the inner and outer cylinders for a given voltage across the gap. r, - 1.711 x 10 m 1  . .e =8.854 x 1 0 ~ o  _2  r a  • 1.903 x 10~ m  3  _3 • 1.08 x 10 cc/cc/°c  VOLTAGE ACROSS CYLINDERS kv(rms)  farads/m  p = 937.7 kg/m  2  2  12  3  , e . „ -, - , ~ / o ( — ) = 3.72 x 10 / c o  E l e c t r i c a l gravity... ... Earth's gravity ' at r=r^  0  n  3  . E l e c t r i c a l gravity Earth's gravity at r=r  y  2  0.2343  0.1703  4  0.9372  0.6812  6  2.1087  1.5327  8  3.7488  2.7248  10  5.8575  4.2574  12  8.4348  6.1307  In Table 1, the values of e l e c t r i c a l gravity compared to earth's gravity are shown both at the surface of the inner cylinder and that of outer cylinder.  At 8.0 kv rias voltage difference  between.the cylinders, the average e l e c t r i c gravity i s three times that of the earth's gravity.  Above this voltage difference, i t  rises even more sharply. .The form of the equations. (50) and (51) and the boundary conditions (46) shows that, i n the marginal state ( i . e . when a=0) , f o r a given value  23 of a, the dimensionless parameters whose values are s u f f i c i e n t to determine uniquely the d i s t r i b u t i o n of U and G are the e l e c t r i c a l Rayleigh number 2.3  R and the gap to inner radius r a t i o  X.  The Exchange .of S t a b i l i t i e s In general o" i s a complex quantity and i s a function of the  physical quantities involved and of the parameters characterizing the p a r t i c u l a r pattern, of the disturbance.  I f Rl(a) i s p o s i t i v e ,  the disturbance increases, but i f i t . i s .negative, the disturbance dies away.  In a.set of possible disturbances, the mode for which  Rl(o)is. a-maximum w i l l , be the f i r s t to appear beyond s t a b i l i t y ,  when  the Rl(o) i s just equal to zero, the l i m i t i n g condition of s t a b i l i t y will.be realized.  I t . i s important  to know.if the Im(a)  i s zero  when the F l ( a ) i s . v I f .Im(a) i s not zero, the disturbance manifests i t s e l f .in the form. of. a wave.motion. and the system i s said to be overstable. -When Im(o) i s zero i f the Rl(a) i s zero, we say that the p r i n c i p l e of exchange o f . s t a b i l i t i e s i s v a l i d . vanishing of .cr  Since the  means that a l l time v a r i a t i o n s disappear, this l i m i t i n g  condition represents a steady state i n which the disturbance just maintains  itself.  To consider, the p r i n c i p l e of exchange of s t a b i l i t i e s f o r the problem i n hand,.it i s . s l i g h t l y .more convenient non-dimensionalized  form of equations  to write the  ( 4 0 ) - ( 4 2 ) with respect to  24 r^,  the radius of the outer cylinder. ^3 r  (DD -a -o )(DD^-a )U = 2  A  2  1  1  < * - l D  D  a  2  1  P 0  l  ) Q  (55)  7  =  (56)  and U = DU = 6 » 0 at r = n where D - ^  Then, we have  ,  D* - ^  and 1.  (57)  + ± ,  (58)  ar a  l =  k r  2  " -T  •  ( 5 9 )  a R, " ~r  and  X  n -  T T  .  1  (61)  2  H  Multiplying equation (55) by rU* (the star superscripts represent complex conjugates) and integrating i n the i n t e r v a l r = n and r = 1 , we get  f  r U * ( D D - a - a ) ( D D - a ) U dr = - R ^ 2  A  2  1  1  A  1  [ in  ^ r  U* dr  (62)  Since U and i t s derivative vanish at the boundaries, the l e f t hand side of equation (62) can be shown to be positive d e f i n i t e (Chandrasekhar, 1961, p297). j  Thus  rU*(DD -a -a )(DD^-a )U dr 2  A  =  f  1  'n  2  1  1  1  r|(DD^- )u| dr ai  2  2  + orf[r|g| Jn  2  + (^ + a ^ r ) ! ^ ] 2  dr  (63)  25 Substituting f o r U* from equation (56.)  i n the r i g h t hand side  of equation (62) , we get  £  ^ * U  dr =  l  f ( *D-a -Pcv*)0*dr n  n  = f  1  (64)  2  - DJ)0*dr - (a +Po*) t r * 1 In 2  1  Jn  L  Again making use of the boundary conditions  r  (65)  dr  on 0 and integrating by  parts, we.can e a s i l y show that  J  1  ^ |D0| dr +  | D^D0*dr = -  2  Combining equations (62), + I  %:>  (63),  (65)  and (66),  2  1  1  2  1  we get (67)  = R a t ( a + P o * ) I + 1^ + Ig]  2  (66)  D9*dr  3  where I  x  I  2  = J  1  [r|-BU| + ( i + 2  j ^ . . r|(DD^-a )u| dr 2  2  a r ) | u | ] dr 2  2  (68)  (69)  26  1  2~t  )  [„  ^lefdr  (70)  i|D6| dr  (71)  2  and I  5  = -2  f  1  Jn  1  0_ D9*dr 2 r  The integrals 1^, !<. i s complex.  (72)  , I^ and 1^ are p o s i t i v e d e f i n i t e  while  Equating the imaginary parts of equation (6 7),  we  obtain Im(a)  [I + R 1  2 i a ; L  PI ] = R a Im(I )  (73)  2  3  1  1  5  and no general conclusions regarding the s t a b i l i t y of flow can be drawn from this equation.  The v a l i d i t y of the exchange of s t a b i l i t i e s  cannot be rigorously proved f o r this problem. ;  In .the next.section, i t i s shown that, for a narrow gap, the s t a b i l i t y equations reduce to the form of the. Rayleigh-Benard case..  The p r i n c i p l e  of the exchange of s t a b i l i t i e s f o r the Rayleigh-  Benard form of equations has been proved elsewhere chapter II).. .In.chapter I I I , the experiments  (Chandrasekhar,  on e l e c t r i c body  force, induced convection are described and they show that the i n s t a b i l i t y appears  t o s e t i n as a s t a t i o n e r y c o a v a c t i v a flew. Without attempting  a.theoretical j u s t i f i c a t i o n of the p r i n c i p l e  of exchange of  s t a b i l i t i e s , we s h a l l assume i t s v a l i d i t y to obtain the numerical solution, of the eigen-value problem.  27 2.4  The equations.for the case of a narrow gap.  If the gap r - r (~d) between the cylinders i s small compared 2, 1 r "t°r to t h e i r mean radius 1 2(=r ) ,D. can be simply replaced by D, — 2 — o * (Chandrasekhar,1961, p 402). In this scheme of approximation, equations (40) and (41), become 0  9  2  9  „ 2  9 Z  (D -a -a)(D - a ) U = -  V ° r~~  \  [ln — j V  ad "vp •  ( 1  2  2 and  (D -a -Pa)G 2  =-  2  < i~ r„ T  T  ,  2.  Kin •  r  where 5 - H!i d  .  = f ,  k  ) d 2  —  JJ  r  l P/v = ^ d  '  3e . 8T p ;  r  ©_ 3 o  (74)  (75)  O (76)  The boundary conditions (42) are U = DU =. 0 = 0 at £=0 and 1  (77)  By making the transformation (T-T.)d • •  -  2  U+ U  1  (78)  icr In — ° l r  equations  (74) and (75) become  ( D - a - o ) ( D - a ) U = -R0  (79)  (D -a -Po)0 - U  (80)  2  2  2  2  2  where  .  2  ,  . K-  (81)  —  VK  28  g  g  and g are now  defined at the mean radius r ^ and have the following  meaning  ^ K - - - r - ^ -  2  ^<H>,  1  <  82)  T -T and  1__2_  =  g  ( g 3 )  iIn  r  r  r  2— l  Equations (79) instability  and (80)  (Chandrasekhar,  are the same as for Rayleigh-Benard  chapter 2 ) „ The p r i n c i p l e of exchange  of s t a b i l i t i e s f o r this case holds and the i n s t a b i l i t y occurs at a Rayleigh number 1707., 762  corresponding to a = 3 o l l 7 „  This analysis serves to show that the p r i n c i p l e of exchange of s t a b i l i t i e s i s . v a l i d i n the small gap approximation.  However, i t  ignores the curvature e f f e c t of the cylinders and cannot be used for any comparision with the experimental r e s u l t s . 2o5  The solution f o r a wide gap when the, marginal state i s stationary, In the marginal state (a=0), the equations (55)  become  _ „ (DD* - a ^ r u  and  (56)  o n (84)  = -R.^^ r  (D*D - a )0  and  (85)  = ^  2  ±  On eliminating U between equations.(84) and (85), Q  VI . 7 V r +  Q  +  5_ _ 2  2  3  r  1  IV _  - — r  - a. 6 1  - -R^  6  + 3a  +  r  . 2  + ( -j r  ^  ) 0  3  ]W_  2  ^r  1 1 4  III  r _  2  - ( -j - — r  we obtain  4  —  ) 0  r (86)  29  with the appropriate boundary conditions  0 = 0  II J  —  1  -  &Q  2  and 1  r  =  1  Q  III . 2  + -p  II  - a  2 1 x  a  0  2 l  = 0  (87)  ft  -0  l  r  at r  I  0  2 Alternately, we can eliminate 0 between equations (84) and (85).  The eigenvalue problem, then., becomes 2 TT  V I  U  I  -4-1^.4.  + -U  1  *o  6  2  + (—-pp •» 3a r  .  +  r  ! ^ .  )  ± f + (^ r  ^ l  2  I V  )U  i  D  >UT  „. Z ! i _  (  +  r  ,3 , - ( — +  2 3 S  r  _ 3_ _  r  2  W.IH —)U  6  )  0  1  - 4...II r — - 3a. )U r  .  2 u_  r  r  ( 8 8 )  r  subject to boundary conditions  U.= U  1  , U  I V +  | u  - (\ + 2a  m  2 1  )U  1 1  + (^ - ^ U  r at r - — 2  1  - ( \ - ^  r  r  -a^U  (89)  r  The c r i t i c a l values of  and a^ were found by solving  numerically equation (86) with boundary conditions (87). The. general solution of ..equation (86) i s 0 = A © 1  1  + A 0  constants.  2  2  0  r  1.  and  5  + A © 3  3  + A 0 4  4  + A © $  5  +A 0 , where A . ^ . . .Ag are 6  6  The condition that these solutions 0 ,0,.,,8, form a 1 z o fundamental set i s that t h e i r Wronskian i s not zero. There i s an  30  i n f i n i t e number of possible sets of solutions.  One simple set pai* be  found by choosing 0^(r) such that 0 (r /r ) 1  I  = 1, © . ^ ( ^ / r , , ) =  2  ©^Vi/O  = ....-6^ {T^TJ  = 0 (90)  and defining 0 (r) where n = 2,3.. <6 as that solution which s a t i s f i e s n the i n i t i a l conditions  0 ^ ( r ) « 0 n  I f m 4 n,ra•-• 1,2...6  0 ~ (r) = 1 n  i f m = m m = 1,2...6  m  1  (91)  (92)  Then 0 , ( r ) , 0_(r).... 0, (r) form a fundamental set and the value 1 2 6 of  t h e i r Wronskiah at r = — 2 !  i s unity (Ihce,. 1956jp 119).  r  For a. given, value of a^ and. R^. the solutions 0^ .(£),, 0^(r) ... 0^(r) , with the i n i t i a l conditions (90)-(92), were- found numerically bf Xhe method, of Runge-Kutta by breaking the s i x t h .order d i f f e r e n t i a l equation (86) into s i x simultaneous  f i r s t Order equations'  (Ford, 1955, chapter 6) by writing <z>'  1  0 e  1 1  0  I V  =• G  m  0*  - H = J - K =  L  and L  1  + ^ -  L +  ( ^ r  o -(^5 r  5a  -  -3 r  2 1  -  3a ) K 2  ( ^  ^  +  *  ) J +  4 7a 2 "-)G + ( R / i _ . )0= 0 r 4 ' r r  ¥  6  a  1  ( ^ r  -  +  3a^)U  r  (93)  31 Knowing 0,(.r), Q (r) 1 2 n  ... 0, (r) o  and thus 0, the boundary  conditions (87) were substituted to give a set of s i x l i n e a r homogeneous equations (three at the inner boundary and three at the outer boundary).,  The condition that, they have a n o n - t r i v i a l  solution i s that the determinant D  E  T  =  l l 12  a  a  a  13  a  14  a  15  a  16  - f(a ,R ) i 0 1  1  a,21 l  31 41 51  a  61  a  62  a  63  (94)  %6  where a,, (j; = I...6) aire the coefficients- of A. i n the f i r s t 1.1 ' 3 J  equation; a^j are c o e f f i c i e n t s o f : i n .the..second,.equation and so on. For  a given.value pf-a^, some .value of  w i l l cause the  determinant to vanish. .To f i n d that value, determinant ( 9 4 ) was calculated f o r .increasing values pf R^ t i l l the determinant changed sign..... Knowing ..positive^ ..and- ,.negat;iy& -values, 'of the determinant, corresponding to two values pf %.^*. & l i n e a r i n t e r p o l a t i o n scheme (regula f a l s i , method, Mathews and Walker, 1964, p339) was used  3.08  3.10  3.12  3.14  DIMENSIONLESS WAVE NUMBER a FIGURE 2. DEPENDENCE O F THE NUMERICALLY DETERMINED ELECTRICAL RAYLEIGH NUMBER ON Ti i'L DI KENS! G,<' LL".r>:. WAVE NUMBER.  33  FIGURE  3-  THE VELOCITY  PROFILE  THE  IS  VELOCITY  AMPLITUDE.  AT  THE ONSET  N O R M A L I Z E D TO  UNIT  OF  CONVECTION.  MAXIMUM  34  FIGURE  4.  THE  TEMPERATURE  CONVECTION. TO  UNIT  THE  MAXIMUM  PROFILE  AT  TEMPERATURE AMPLITUDE.  THE IS  ONSET  OF  NORMALIZED  0.0  FIGURE 5.  0.2  0.4  0.6  o.e  THE STREAMLINES AT THE ONSET OF INSTABILITY. THE STREAM FUNCTION ^ (PROPORTIONAL TO rU cos az) HAS BEEN NORMALIZED TO UNITY AND THE CELL PATTERN IS DRAWN SYMMETRICALLY ABOUT z=0. THE UNIT OF LENGTH IS THE THICKNESS OF THE ANNULUS.  1.0  36 to i t e r a t e and f i n d  for which the determinant vanished.  R^, i n  general, converged to a f i n i t e value (to eight s i g n i f i c a n t figures) within s i x i t e r a t i o n s .  For these values of a^ and R^,  the value  -12 of the determinant was less than 10  and was considered adequate.  Knowing a^ and R^, the corresponding values of a and R were found by making use of equations (59) and (61).  The results of these  computations are shown i n figure 2. The solution of i n t e r e s t i s the minimum value of R for which f(a,R) = 0.  Quadratic i t e r a t i o n was used here to f i n d the  value of a f o r which R i s minimum. by McFadden(1969). are R  c  The method has been described  The c r i t i c a l values of R^ and  = 2119.346 ;.  a  c  =  found numerically  3.117  using the values r ^ = 1.711  cms.;  = 1.903  cms.  for the r a d i i of the cylinders. These values were also calculated by solving the s i x t h order equation i n U and were found to agree within three decimal places (seven s i g n i f i c a n t f i g u r e s ) . Figure 3 shows the p r o f i l e of the v e l o c i t y d i s t r i b u t i o n corresponding to L distribution.  and a^.  Figure 4 shows the temperature  The stream l i n e s at the onset of i n s t a b i l i t y are  shown i n figure 5.  37 CHAPTER I I I EXPERIMENTAL ARRANGEMENT AND RESULTS  3.1 Introduction Chandra(1969) has described the c y l i n d r i c a l experimental arrangement i n d e t a i l . He also presented some preliminary results oh e l e c t r i c a l body force driven convection. Although the heat transfer measurements c l e a r l y indicated the change from conduction to convection regime, the measured Nusselt number ( r a t i o of t o t a l heat transfer to heat transfer by conduction alone) In the conduction regime was found to be 10-15% less than i t s value of unity. No d e f i n i t e explanation f o r this discrepancy could be given. However, i t was suspected that the discrepancy arose p a r t l y due to the heat losses from the ends of the heating c o i l and p a r t l y due to the errors i n the inference of the temperature of the outer cylinder. The outer cylinder was cooled by transformer o i l kept at a constant temperature by c i r c u l a t i n g i t through an external thermostatic bath. The temperatures at the i n l e t and outlet points (figure 6, 11 and 12) to the thermal f l a s k were measured and t h e i r mean was  taken to be  the temperature of the outer cylinder. However, f o r a short contact time of flowing l i q u i d at a temperature lower than that of the s o l i d w a l l , the f l u i d temperature changes appreciably only i n the immediate V i c i n i t y of the w a l l (Bird, Stewart and Lightfoot, 1966, p349). Away  38 from the. w a l l , there i s hardly any change i n f l u i d temperature even when there i s increased heat transfer between the cylinders. It was therefore, f e l t necessary to do the experiments again to be able to compare experimental results with theoretical values.  In.these experiments, the heating c o i l was placed much  closer to the wall of the inner cylinder to minimize heat loss from the ends and to assure r a d i a l heat transfer.  However, heat  transfer and temperature measurements showed that, this could not account f o r the observed discrepancy. Next, temperature of the outer cylinder was measured at zero applied p o t e n t i a l independent of. the i n l e t and outlet temperatures. The true temperature of the outer cylinder was found to be d i f f e r e n t from the. mean.  A.linear r e l a t i o n was obtained between the mean  of the i n l e t and outlet temperatures and the true temperature of the outer cylinder-  This was done only when there was no p o t e n t i a l  difference across the cavity.  When the outer cylinder was at a high  voltage,.Its temperature could not be.measured d i r e c t l y .  However,  from the l i n e a r relationship, the temperature of the outer cylinder was found i n d i r e c t l y under the assumption that the flow and temperature d i s t r i b u t i o n i n the thermal flask are not altered by the high voltage.  The Nusselt number now gave a value of unity i n the  conduction regime.  39  c=i STAINLESS STEEL  ^  .  .  .  .  FIGURE 6. A CROSS SECTION OF THE APPARATUS..  ^  40 3. 2  Experimental Arrangement The apparatus,  detailed in.figure 6, consisted of two  coaxial cylinders (1 and 2) of stainless steel-  The cylinders  were polished to a mirror f i n i s h to avoid any sharp points which might cause e l e c t r i c breakdown at high voltages. The inner cylinder (2) was,constructed i n two parts.  The  central part was made.from brass tubing i n which holes at 8, 9 and 10 were d r i l l e d to accept thermistors,,  A tight f i t t i n g  s t a i n l e s s s t e e l tubing was then forced over the brass tubing.  End  pieces were machined from stainless s t e e l and f i t t e d to the central part with inside corners at the lines of contact to prevent the occurrence  of corona.  The heating c o i l . ( 7 ) , made of a close wound  30 S.W.G. diamel coated'Mancoloy 10' wire on a f i b r e g l a s s rod, was a c l o s e . f i t inside the inner cylinder.  After i n s t a l l i n g  calibrated thermistors i n the receptacles at.8, 9.and 10, the. heating c o i l was centered, and the remaining  space inside the inner  cylinder was f i l l e d with thermal epoxy and allowed to set.  Besides  the end points, the heating c o i l was also tapped at two intermediate points d i r e c t l y opposite the thermistors at 8 and 10.  The leads f o r  the thermistors and the heating c o i l were brought out the top of the cylinders. The inner cylinder was located coaxially inside the outer cylinder (1.) by means of plexiglass centering devices at each end. The locations of the centering devices were well beyond the measurement length between the points 8 and 10.  The outer cylinder was.  equipped with t o r o i d a l terminations to prevent  the occurrence of  — G > POWER SUPPLY  R V W 2  4  42 corona.  Table 2 summarizes the important dimensions of the completed  cell. Table 2 Cell  Dimensions.  Radius of the inner cylinder Inner radius ,= 1.455 ± 0.002 cms. Outer radius = 1.711 ± 0.002 cms. Radius of the .outer cylinder Inner radius = 1.903 + 0.002 cms. Outer radius = 2.114 ± 0..002 cms. Length of the measurement section between the points 8 and 10 L = 8.25 ± 0.02 cms. The assembled  apparatus was set i n a double walled thermal  flask (5) f i l l e d with transformer o i l .  This o i l was circulated  through an external thermostat, thus enabling the temperature of the outer cylinder to be maintained constant.  The temperature i n  the thermal flask could be controlled to ±0.02°C. The present.setup didnot allow to observe f l u i d motions visually.  The onset of thermal convection was detected by making  heat transfer and temperature measurements. The Heating C i r c u i t : shown i n figure 7.  The control c i r c u i t f o r the heating c o i l i s As already mentioned, the heater was a three  sectioned heating element.  Currents i n the three sections were  adjusted by means of the rheostats R^ and at  u n t i l the thermistors  8, 9 and 10 gave the same.temperature reading (control to  about 0.01°C was found feasible) to ensure uniformity of.temperature  43 on the surface .of .the inner cylinder.  At equilibrium the  heat transfer occurring i n the..section.between  the points 8 and  10 was taken to be equal to the dissipation of e l e c t r i c a l energy i n the central section of the c o i l . the voltmeter  The ammeters 1^ and  and  were a l l of .1% accuracy (made by Weston, Model 81).  The current f o r the heating element was drawn from a d i r e c t current, (Hewlett Packard, model 6290A) power supply.  The power  input i n the central, section .of the c o i l . i s . given by (i-j..  - i )v 2  1  watts  The Thermistors: The glass coated thermistors of .043 i n . diameter had a nominal resistance of 50,000 ± 20% ohms at 25°C and a dissipation.constant.in a i r of 0.7 milliwatts per °C. A l l thermistors were calibrated before use.  The c a l i b r a t i o n was per-  formed against a.N..P.L..calibrated platinum resistance thermometer. The.calibration points were recorded over a 15-28°C i n t e r v a l . The results were analyzed,by  the. method .of. least squares f o r f i t  .to. the. orthogonal, polynomials f o r the function lnR =.A + B/T + C/T  2  where ,R .is... the. resistance of the thermistor at temperature T°C and A,- B...and. C. are .the. constants, of .the .cubic equation. .The results of c a l i b r a t i o n and the difference between the calibration, resistance and. the...computed resistance. are l i s t e d i n tables 3 and 4 (Appendix 1).  -o-  -O  o  Sj  = 29992.3  O H M S  S2  = 29992.8  O H M S  R  DECADE  T, , T  2  FIGURE 8. THE BRIDGE CIRCUIT.  ETC.  RESISTOR THERMISTORS  45  •  I  _J  I  1  15 25 35 TEMPERATURE IN DEGREES CENTIGRADE FIGURE 9.  KINEMATIC VISCOSITY AS A FUNCTION OF TEMPERATURE.  47 The resistance of the thermistors was  read with a d i r e c t  reading bridge s p e c i a l l y designed f o r this purpose (figure 8). The High Voltage Supply:  In a l l experiments, the inner cylinder  was grounded and the outer cylinder was kept at a high alternating voltage.  The 60 Hz high voltage was supplied from a step-up  transformer made by the Universal Voltronics Corporation. voltage s t a b i l i z e r was the transformer.  A  inserted between the mains and the input to  This eliminated the fluctuations of the power  l i n e and gave a constant output voltage. A high accuracy (1%) e l e c t r o s t a t i c voltmeter was used to measure the applied p o t e n t i a l difference between the cylinders, The D i e l e c t r i c Liquid:  The d i e l e c t r i c l i q u i d for these  experiments  was DC200 e l e c t r i c a l grade s i l i c o n e o i l (supplied by Dow whose physical.properties are l i s t e d i n table 5.  Corning)  The s i l i c o n e o i l  was selected because i t has a high e l e c t r i c strength and i s available commercially i n any desired v i s c o s i t y grade.  The dynamic  v i s c o s i t y and the d i e l e c t r i c constant of the o i l as a function of temperature were measured using a f a l l i n g b a l l viscometer and a standard capacitive c e l l respectively.  In figure 9, the kinematic  v i s c o s i t y of the o i l i s plotted as a function of  temperature.  Figure 10 shows the v a r i a t i o n of d i e l e c t r i c constant with Other properties l i s t e d in.table 5 are taken from Dow B u l l e t i n 05-213 of July  1968.  temperature.  Corning  48 Table 5 Physical Properties of Silicone O i l * Density,. p--0.9377 gm. cm  -3  Coefficient: of volume expansion,  a- 1.08x10  -4 cal. Thermal conductivity, x = 3.2x10  cm  -1  sec  S p e c i f i c heat,. Cp= 0.43 c a l . gm. D i e l e c t r i c constant,  K =2.64 e  -1 Kinematic viscosity,.v = 12.225 cm sec. *A11 properties are given at 23°C. 3.3  Experimental Procedure : The c y l i n d r i c a l annulus, previously cleaned, was f i l l e d  with degassed e l e c t r i c grade s i l i c o n e o i l .  The cavity was.further  connected to a vacuum pump to remove any tiny bubbles s t i c k i n g to the walls of the cylinders.  Having made a l l e l e c t r i c a l connections,  the external thermostat was switched on. . In a l l experiments, the temperature, of the outer cylinder, T^, was kept lower than the temperature of the inner cylinder, T » An i n i t i a l run, to correlate the temperature of the outer cylinder, T^> with the mean of the i n l e t and outlet temperatures, was done without any high voltage across the cylinders.  For t h i s ,  two thermistors were attached to the wall of the outer cylinder. The inner cylinder was heated and when the.conditions became stable, the temperatures at points 8, 9 and 10 (figure 6) were measured. The currents i n the three sections of the heating c o i l were adjusted  TEMPERATURE CORRECTION IN CENTIGRADE DEGREES -n  <r> 70  rn I—  2 o  TI  — z rn H  1  tn z o ro o m O  i o c —i r~ rn  —I  —i m  -6  u rn  m m  -i 5 rn  H  >  o 70 -Tl  <?  c  —I rn  m  X  m >  o c  —I rn 70  o -< LO  z  rn o 70  m  70  r~  >  —I m  70  m tn tn  6<7  50 with rheostats R^ and  (figure 7) u n t i l a l l the temperature  readings on the surface of the inner cylinder were within 0„01°G. The temperature of the outer cylinder and that of the i n l e t and outlet points,were a l l noted. After 10 minutes, a l l temperatures were measured again and i f there was.no change then the readings were accepted.  The procedure was repeated f o r several values of  heat transfer.rate across the. cavity.. I t was found that the actual temperature of the outer cylinder less the mean temperature read by the i n l e t and outlet thermistors (11 and ,12 in. figure 6) i s l i n e a r i n the heat transfer.rate between the. cylinders as shown i n figure 11. . , When the. heat transfer, and.temperature measurements were done with the outer cylinder, at.a.high alternating p o t e n t i a l , i t was not possible to .measure its.temperature. transfer .in .the circulated..transformer 1  Since the flow and heat o i l f i l l i n g the thermal flask  . .iare. l i k e l y , to.be,'largely, unaffected, by the ..potential of the outer cylinder, :itwasi assumed that .this, i s precisely so. ;  Knowing the  . heat transf er..,rate..and. the mean, of ..inlet-outlet temperatures, the plot, shown -in, figure. 11 provided, an i n d i r e c t means of obtaining the temperature of, .the, outer cylinder. ...The experiments.were run for.the. following values of p o t e n t i a l difference across the gap. 0 kv rms,. A.06 kv rms, 6.00 kv- rms, 6-92 kv rms, .7.91 kv rms, 9.63 kv rms, 10.15 kv . rms,.. 10.80 kv rms.  -1  X  IN M A  100  MA  RANGE  FIGURE 12. DIFFERENCE BETWEEN THE READING OF AMMETER ^ A N D LABORATORY STANDARD AS A FUNCTION OF THE AMMETER READING FOR DIFFERENT CURRENT RANGES. THE CURRENT DIFFERENCE IS PLOTTED AS Y AND THE AMMETER READING AS X.  Y IN MA  X  IN MA 100  MA RANGE  300 X  IN MA  300 MA RANGE  +10-  0.6 X IN AMP 1 A RANGE  -10  FIGURE 13. DIFFERENCE BETWEEN THE READING OF AMMETER I AND LABORATORY STANDARD AS A FUNCTION OF THE AMMETER READING FOR DIFFERENT CURRENT RANGES. THE CURRENT DIFFERENCE IS PLOTTED AS Y AND THE AMMETER READING AS X . 2  Y IN VOLTS +.05 -  X  -05  IN  VOLTS 2.5 V RANGF  +.05  -.05-  X  IN VOLTS 5.0 V RANGE  X  IN  VOLTS  10 10.0 V RANGE  FIGURE 14. DIFFERENCE BETWEEN THE READING OF THE VOLTMETER V , AND LABORATORY STANDARD AS A FUNCTION OF THE VOLTMETER READING FOR DIFFERENT VOLTAGE RANGES. THE VOLTAGE DIFFERENCE IS PLOTTED AS Y AND THE VOLTMETER READING AS X.  54 A l l the observed data i s tabulated i n tables 6-13 (Appendix I I ) . 3.4  Errors and corrections. The ammeters 1^ and I^ and the voltmeter  (figure 7)  were calibrated against a laboratory standard of 0,1% accuracy. results of c a l i b r a t i o n are shown i n figures 12-14.  A l l values of  currents and voltages were reduced using these graphs. values of i , i  and v  The  The corrected  and the average temperatures of the inner  and outer cylinders are tabulated i n tables. 14 through 21 (Appendix I I I ) . :Errors-,..due..-to. s e l f heating of ..the, thermistors:  The thermistors were  s p e c i f i e d to have a. dissipation, constant i n air, of..0.7 m i l l i w a t t per °C. Accordingly, a, .power, of .0.7 .milliwatt in..the thermistor w i l l change its- temperature by. 1°C which w i l l i n turn change i t s resistance, by about 5%.  This i s referred to .as ..a self-heating error. I t  can be reduced by.,properly, .selecting the bridge voltage. ,.  , The.: self-heat effect, was. restricted, to less than the 0<01°C  allowable precision..,, Taking, a.reasonable .amount of error due to selfr-heating-to,.be.,50%,,o£,. allowable p r e c i s i o n or 0..005°G, the max-6 imum..power.„al.lowed. through, the. thermistor, i s . 0.7 x 0.005 = 3.5 x 10 wat Taking an average value f o r the resistance of the thermistor to be 50,000fi, we get f o r the voltage across the thermistor a —6 value  3.5x10  3 x50xl0  or 0.42 v o l t s .  of 0.84 volts across the bridge.  We can, therefore, use a maximum  1  2  3  4  5  HEAT TRANSFER RATE IN WATTS FIGURE 15.  DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 0 KV RMS BETWEEN THE CYLINDERS.  HEAT TRANSFER RATE IN WATTS F1GURE*16. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 4.06 KV RMS BETWEEN THE CYLINDERS.  FIGURE 17.  DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 6.00 KV RMS BETWEEN THE CYLINDERS.  LU  O Z  GO LU LU LU  cr (3cr LU LL. LU LL- Q  1  4 HEAT TRANSFER RATE IN WATTS  FIGURE 18. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEA'i RATE FOR 6.92 KV RMS BETWEEN THE CYLINDERS.  iR^-S"  FIGURE 19. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 7 . 9 1 KV RMS BETWEEN THE CYLINDERS.  LU  1  2  3  4  5  6  HEAT TRANSFER RATE IN WATTS FIGURE 2 0 . DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 9 . 6 3 KV RMS BETWEEN THE CYLINDERS.  8  HEAT TRANSFER RATE IN WATTS FIGURE 2 1 . DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 10.15 KV RMS BETWEEN THE CYLINDERS.  —  LU  O  CO  z:  LLJ  U J  LJJ  cr  cr  o  LU U_ U_  LU Q  Q  LU  LU  or 3  UJ  L u  /  —  <  10.80 KV  cr  —  jfj  z  2  —  J.  1 1  I 2  1 3  1 4  1 5  1 6  HEAT TRANSFER RATE IN WATTS FIGURE 2 2 . DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 10.80 KV RMS BETWEEN THE CYLINDERS.  63 A bridge voltage of 0.5 volts was used for the resistance measurements which reduced the error due to self-heating f a r below the required p r e c i s i o n . Error due to f i n i t e thickness., of .walls of the cylinders: of the inner cylinder was measured by the thermistors the inner wall„;  The temperature  mounted on  The temperatures of the outer cylinder were those  at the outer surface.  For any true calculation of conductive heat  transfer through the l i q u i d , i t i s e s s e n t i a l to correct for the f i n i t e w a l l thicknesses...  The thicknesses of the walls were s u f f i c i e n t l y  small that i t was estimated that the maximum error introduced i s less than 0.5% and i t was n o t considered necessary to make this correction. Errors due to .internal resistance of 1^, 1^ and V^: The s e n s i t i v i t y of the voltmeter. V^, was 20,0000/volt. was  The resistance of.the ammeters  0.10 on 1A range compared to about'130.for the mid-section of the  heating c o i l . . I t was,  therefore, found unnecessary to do any corrections  for^the. i n t e r n a l resistance, of the meters. .3.5  Experimental Results. Experimental results, i n the form.of.the temperature difference  between.the cylinders .plotted against the heat .transfer rate i n the central section of the annulus are shown i n figures 15 through .22 for the range of voltages applied to the. outer, cylinder. For potentials up.;to 7.91 kv rms, the relations are l i n e a r , i n d i c a t i n g  & 9.63 KV 1.3 _  1.2  o 10.15 KV © 10.80 KV  ©  —  1.1  © O  &  1.0  O O  ©  ©  ©  I  1  i  1  1  1  1  500  1000  1500  2000  2500  3000  35.00  ELECTRICAL RAYLEIGH N U M B E R  FIGURE 2 3 . NUSSELT NUMBER AS A FUNCTION OF ELECTRICAL RAYLEIGH NUMBER.  65  conductive heat transfer even f o r the maximum temperature For potentials of 9 . 6 3 ,  attained.  differences  1 0 . 1 5 and 1 0 . 8 0 kv rms,  distinct  breaks occur i n the l i n e a r relationship i n d i c a t i n g the onset of convective heat transfer. A l l the results are summarized i n figure 2 3 as a plot of the Nusselt number v (l 1  W  - 1)  1  4.1854  U  ln(r / )  2  2  [2TT L(T X  1  -  r i  T,,)]  against the e l e c t r i c a l Rayleigh number, R (equation 5 2 , Chapter I I ) . In the above equation x i s the thermal conductivity of the f l u i d and L i s the length of the measurement section between the points 8 and 1 0 (see figure 6 ) . The values of v i s c o s i t y and d i e l e c t r i c constant entering i n the expression f o r R were taken at the mean of the temperatures  of the cylinders.  so s l i g h t l y on the . temperature  The thermal conductivity depends that i t s v a r i a t i o n with  temperature  Other constants used f o r the o i l , a r e l i s t e d i n Table 5 .  was ignored.  Figure 2 3 shows the experimentally determined  critical  e l e c t r i c a l Rayleigh number for the onset of i n s t a b i l i t y to be R  3.6  c  =  2200  ±  100.  Accuracy The accuracy of the f i n a l results f o r Nu and R i s estimated  to be,±5% when a l l the factors are considered.  66 CHAPTER IV DISCUSSIONS AND. CONCLUSIONS 4.1  Discussion of Results The figures 15 to 22 show the plots of the heat transfer rate  against the temperature difference between the cylinders f o r a range of voltages applied to the outer cylinder,,  For lower voltages the plots  are l i n e a r , i n d i c a t i n g the heat transfer i s by conduction alone.  Above  7.91 kv rms, however, the plots show d i s t i n c t breaks which occur at the onset of i n s t a b i l i t y  (Schmidt and Milverton, 1935).  The e l e c t r i c a l  buoyancy force f o r small voltages was not s u f f i c i e n t to overcome thermal and viscous dissipation even f o r the maximum temperature differences created across the gap.  However, i f the temperature of the inner  cylinder i s raised further, we should expect appreciable convective heat transfer even f o r those voltages. The plot of the Nusselt number against the e l e c t r i c a l Rayleigh number i s shown i n figure 23.  In the conduction regime, the Nusselt  number i s found to have a s l i g h t downward trend from the expected of unity.  value  I f there i s a small temperature drop at the outer w a l l  contact point during the i n i t i a l run with no high voltage, this could lead to an underestimation of the temperature difference of figure 11. Consequently, the impressed would be overestimated.  temperature difference between the cylinders  A l i n e a r increase of this contact temperature  difference with impressed  temperature difference could r e s u l t i n a small  decrease i n the Nusselt number with increasing Rayleigh number. The value of the e l e c t r i c a l Rayleigh number at which the onset of i n s t a b i l i t y occurs i s :  R  = 2200 ± 100. c The t h e o r e t i c a l l y determined value of R  i s 2.119.346 and compares  67 w e l l with the experimental value.  The disturbances which w i l l be mani-  fested at marginal s t a b i l i t y are characterized by the wave length A = ^ k  = ^=2.02d. a c  The value of a^ f o r this case i s same as f o r the convection (a =3.117). c  In the Rayleigh-Benard  Rayleigh-Benard  i n s t a b i l i t y , the Nusselt  number i s a function of the Rayleigh number only.  Here, however, the  Nusselt number depends on the e l e c t r i c a l Rayleigh number and the r a t i o of the inner cylinder radius to gap width. 4.2  Conclusions The present investigations have demonstrated the f e a s i b i l i t y of  constructing a laboratory model of thermal convection to simulate large scale geophysical flows that occur under a (nearly) central g r a v i t a t i o n a l field.  The marginal s t a b i l i t y analysis of the flow was  several simplifying assumptions.  done by making  The general v a l i d i t y of the p r i n c i p l e  of exchange of s t a b i l i t i e s could not.be proved f o r this problem.  However,  the q u a l i t y of the agreement of the experimental results with the numerically determined value (assuming the p r i n c i p l e of exchange of s t a b i l i t i e s ) i s a strong v e r i f i c a t i o n of the v a l i d i t y of the analysis. In constructing laboratory models of geophysical flows, we need to proceed to spherical geometry and rotation.  We should also be  interested i n the actual form of the flow pattern as well as the values of various parameters at the onset of convection. the development of some new possible-  A l l this w i l l require  techniques, but i n p r i n c i p l e i t now  appears  68 REFERENCES Avsec, D. and Luntz^ M. 1937, E l e c t r i c i t e Et Hydrodynamique - Quelques formes nouvelles des tourbillons electroconvections, C. R. Acad. S c i . , P a r i s , 204, 757. Batchelor, G. K., 1954, Heat transfer by free convection across a closed cavity between v e r t i c a l boundaries at d i f f e r e n t temperatures, Quart. Appl. Math., 12, 209-33. Be*nard, H.., 1900, Les tourbillons c e l l u l a i r e s dans une nappe l i q u i d e , Revue generale des science pures et aplique*es, 11, 126L-71 and 1309-28. B i r d , R. B., Stewart, W.B. and Lightfoot, E. N., 1966, Transport phenomena, John Wiley, New York. Bullard, E. C. and Gellman,. H., 1954, Homogeneous dynamos and t e r r e s t r i a l magnetism, P h i l , trans. Roy. S o c , Sr A, 247, 2 1 3 Chandra, B., 1969, Thermal convection under an a x i a l l y symmetrical force f i e l d , M. Sc. Thesis, University of Western Ontario, London, Canada. Chandra, K. , 19.38, I n s t a b i l i t y of f l u i d s heated from below, Proc. Roy. Soc. (London), Sr A, 164, 231-42. i...Chandrasekhar, S., 1961, Hydrodynamics and hydromagnetic Oxford Univ. Press.  stability,  Debye, P., 1929, Polar molecules, Chemical catalog company. New , York, p27. Ford, L. R., 1955, D i f f e r e n t i a l equations, McGraw H i l l . F u l t z , D.., 1961,. . Developments, i n controlled experiments on large scale. geophysical, problems,. Advances..in .Geophysics,, edited .by H. E.. .Landsberg, 7, 1-103. Greenspan, H. E.,. The.theory of rotating f l u i d s , Cambridge. Gross, M. J . , 1967, Laboratory analogies for convection problems, Mantles of .the earth, and t e r r e s t r i a l planets, edited by S. K. -Runcorn, .Interscience, 499-503.  69 Gross, M. J . and Porter, J . E., 1966, E l e c t r i c a l l y induced convection i n d i e l e c t r i c l i q u i d s , Nature, no. 5058, 1343-45. Hadley, G., 1735, Concerning the cause of the general trade winds, Reprinted i n the Mechanics of the Earth's atmosphere, Smithsonian Inst. Misc. C o l l . , 51, no. 4, 1910. Ince, E. L., 1956, Ordinary d i f f e r e n t i a l equations, Dover Publications. J e f f r e y s , H., 1926, The i n s t a b i l i t y of a layer of f l u i d heated below, P h i l . Mag.., A, 2, 833-44. Kronig, R. and Schwarz, N., 1949, On the theory of heat transfer from a wire i n an e l e c t r i c f i e l d , Appl. S c i . Res., A l , 35-45, Landau, L. D. and L i f s h i t z , E» M., 1960, Electrodynamics of continuous media,.Pergamon press, London. Long, R. R.. (editor), 1955, F l u i d models.in geophysics, Proc. 1st . symposium on the use of. models i n geophysical f l u i d dynamics, Bait. Low, A. R., 1929, On the c r i t e r i o n f o r the s t a b i l i t y of a layer of viscous f l u i d heated below, Proc. Roy. S o c , A, 125, 180-95. Malkus, W. V. R. and Veronis, G., 1961, Surface electroconvection, Phys. F l u i d s , 4, 13-23. Mathews, J . and Walker, R. L.,.1964, Mathematical methods i n Physics, W. A. Benjamin, Inc., New York. Melcher, J . R. and Taylor, G. I., 1969, Electrohydrodynamics: A review of the role of i n t e r f a c i a l shear stresses, Annual Review of F l u i d Mechanics edited by W. R. Sears and M. Van Dyke, 1, 111-47. McFadden, C. P., 1969, The e f f e c t of region of low v i s c o s i t y on thermal convection i n the Earth's mantle, Ph. D. thesis, University of Western Ontario, London, Canada. Pellew, A. and Southwell, R . V., 1940, On maintained convective motion i n a f l u i d heated from below, Proc. Roy. Soc,, A, 176, 312-43.  70 Rayieigh, Lord, 1916, On convective currents i n a h o r i z o n t a l layer of f l u i d when the higher temperature i s on the underside, P h i l . Mag., 6, 529-46. Rossby, C. G., 1941, The s c i e n t i f i c basis of modern meteorology, Yearbook of A g r i c u l t u r e , U. S. Dept. of Agriculture, 599-655. Schmidt, R. J . and Milverton, S. W*, 1935, On the i n s t a b i l i t y of a f l u i d when heated from below, Proc. Roy. Soc. (London), A, 152, 586-94. Silveston, P. L., 1958, Warmedurchgang i n Waagerechten FlUssigk-eitsschichten, part 1, Forsch Ing. Wes., 24, 29-32 and 59-69. Smylie, D. E . , 1966, Thermal convection i n d i e l e c t r i c l i q u i d s and modelling i n geophysical fluid-dynamics, Earth and Planetary Science L e t t e r s , 1, 339-40. Turnbull, R. J i , 1 9 6 8 , Electroconvective I n s t a b i l i t y with a s t a b i l i z i n g temperature gradient, Phys. F l u i d s , 11, 2588-96 and 2597-2603. Vening-Meinesz, F. .A., 1962, Thermal convection i n the Earth's mantle, Continental D r i f t . e d i t e d by S. K. Runcorn, Academic Press. Vest, C. M. and Arpaci, V. S. , .1969, S t a b i l i t y of natural convection i n a v e r t i c a l s l o t , . J . F l u i d Mech., 36, 1-16.  71  APPENDIX I  TABLES OF THERMISTOR CALIBRATION  72 TABLE  NO.  3  TABLE INNER  GF CAL I B R A T I C N POINTS FOR T H E R M I S T O R S i CYLIND E R . ( 3) (2) ( 1 ) T E M P E R A T U R E R E S I S T A N C E RES I S T A N C E R E S I STANCE OHMS OH N S CHMS DEGREES C 72127 7808a 15.037 79S12 68559 6351 7 70364 1 7 . 7 14 60155 55902 20.450 61923 49352 52952 54 6 71 2 3 . 166 43625 46668 48310 2 5 . 8<58 38614 41194 42779 28.646 D I F F E R E N C E BETWEEN C A L I B R A T E D R E S I S T A N C E AND COMPUTED R E S I S T A N C E . ( 3) ( 2) ( 1) R E S ISTANCE RES I S T A N C E RES IS T ANC E TEMPERATURE CHMS OHMS DEGREES C OHMS 0.25 1.13 15.C37 - 0 . 81 1.40 0 . 2 5 G . 8 8 . 17.714 0 . 9 6 0 .86 20.450 0.67 2 . 9 8 1 .39 23.166 6 . 42 0 . 24 2 . 5 1 25.898 -7.42 0 . 7 4 0 . 3 6 4 . 1 0 28.646 ( 1)THE RMIS TOR AT POINT ( 2 ) T H E R M I S T 0 * AT P O I N T ( 3 J T H E R M I S T C R AT POINT  8 ON THE INNER C Y L I N D E R . 9 CN THE I N N E R C Y L I N D E R . 1C ON THE INNER C Y L I N D E R .  73 ABLE N TABLE POINT  CF C L I 3 4 A T I C.\ THE MV. I S T C W S .  TEMP£ RATI ;KE DEGR EES C 11 . 9 14 13 . C 68 15. 8 2 4 1 7 . 742 19 . 6 7 7 ?1 . 6 2 4  < r  ;  FCIMS  FOR  (1) T HE RMIS TOR AT P C I N T THE THERMAL F L A S K . (2) T H E R K I S T G K AT FC I NT THE THERMAL F L A S K .  \  AND CUT L E I  (2 i RES IS VANCE CH.MS 7 9 097 7 18 16 6 52 S3 59 505 54242 49 4 66  (1 ) S I S T i\C E CHrlS 9C481 a ? 16 5 7 4 69 7 6 81 C4 6.? 08 2 56632  D I F F E R E N C E . BETWEEN C A L I B R A T E D COMPUTED R E S I S T A N C E . (1 ) RESISTANCE TEMPERATURE OHMS DEGREES C 3.4 4 1 1 . 9 14 1 3 .868 - 6 , 6.1 15.024 -0, 44 17.742 7. Cc 19 . 6 7 7 - 4 , 19 21 . 6 2 4 0, 8,3  INLET  RESISTANCE  AND  (2) RES I S T A N C E OHMS 3 . 69 -5.25 2.0 6 2.70 - 0 .46 - 0 .53  11 AT  THE  INLET  AT  THE  OUTLET P O I N T  12  POINT  TC TO  74  APPENDIX I I  TABLES OF HEAT TRANSFER MEASUREMENTS  75  TABLE NO. "6 VOLTAGE BETWEEN  THE C YLINDER S = {J )  0.0 KV (6)  (7)  (8)  (4)  (5)  5  16 .61  16.82  0.0  0.0  0.0  17 . 57  17. 57  16.61  16. 78  0. 263  0. 073  2.57  18 .02  18.01  18 .00  16 . 59  16.73  0.304  0.084  2.9 8  13. A9  18. 50  IB. 49  16. 61  16.80  0.349  0.096  3.4?  19.29  19.28  19 . 28  16.62  16.82  0.402  0 . 108  3. 98  20. Cl  2 0. 00  2C. 00  16. 63  16.78  0.446  0.118  4 .45  20.89  20.88  20. 8 3  16. 62  16. 78  0. 496  0. 132  4. 95  21.91  21.90  21 . 89  16.64  16 .79  D .543  0. 138  5.48  22.98  22.97  22. 97  16. 60  16. 74  0.596  0. 154  6. 02  24.13  24.13  24 . 13  16 .60  16.74  0.643  0. 162  6.55  25.4 5  25.45  25. 4 5  16.-73  16.83  0.698  0.177  7.05  26.78  26.77  26. 78  16.65  16.79  C.749  C. 189  7. 58  27. 82  2 7. 82  2 7 . 83  16 .64  16.80  0.787  0.201  7.97  (1)  (2)  16. 34  1 6.35  16. 3  17.58  Q ) T EMPERATURE 0 F INNER CYLINCER AT POINT 10. (2) TEMPERATURE CF INNER CYLINDER AT POINT 9. (3) TEMPERATURE CF INNER CYLINDER AT POINT 0. (4) TEMPERATURE CF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE CF CIRCULATING OIL AT OUTLET POINT 12. (6) CURRENT I I IN AMPERES. (7) . CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VOLTS. :  76  TABLE NG. VOLTAGE BETWEEN THE CYLINDERS  4.06KV  (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  17.88  17.89  17.88  16.65  16.85  0.303  0.088  2.90  16.80  0. 353  0. 113  3.50  16.86  0.402  0. 124  3.74  16.83  0.450  0. 141  4 .18  16.89  0.502  Q. 160  4. 65  16.86  0.547  0. 174  5.33  16. 91  0. 594  0. 189  5.50  17.11  0.644  0. 202  5.96  17.06  0.700  0.2 2 0  6 .50  16.92  0.750  0. 236  6. 95  16.98  0.301  0. 255  7.41  17. 01  0. 845  0. 268  7.81  17.02  0.898  0. 286  8. 37  25 .5 1  29.33  (1) (2) (3) (4) {5 ) (6) (7) (8)  v  3  7  29.33  29.34  16.78  TEMPERATURE OF INNER CYLINDER AT POINT 10. TEMPERATURE CF INNER CYLINCER AT POINT 9. TEMPERATURE OF INNER CYLINDER AT PCINT 8 . TEMPERATURE OF CIRCULATING OIL AT INLET POINT 11. TEMPERATURE OF CIRCULATING OIL AT OUTLET POINT 12. CURRENT II IN AMPERES. CURRENT 12 IN AMPERES. VOLTAGE VI IN VOLTS.  77  TABLE VOLTAGE  BETWEEN T H E  NO.  CYL INDERS=  B 6 .ODKtf  (i)  ( 2)  (3)  (4)  (5)  (6)  (7)  19.49  19.50  19 . 5 0  16.86  17.08  0.437  0 . 139  4 . 03  20. 4 4  20.45  20.45  1 6 . 80  17.02  3.496  0.156  4.61  21 . 3 6  21.37  21.37  1 6 . 87  1 7 . 15  .0.552  0. 174  5.13  2 2 . 35  22.35  22 . 3 4  16.8 7  17.16  0.602  0 . 189  5.58  2 3 . 26  23.26  2 3.25  1 6 . 89  17.21  0. 649  0.204  6.02  24.30  24.29  2 4 .3 0  16.89  17.19  0.701  0.218  6 . 54  2 5 . 54  25.55  2 5 . 53  1 6 . 84  17.12  0.763  0.243  7 .04  26.78  26.79  26.77  16.85  17.12  0.8C1  0.253  7.45  27.91  27.90  27.90  16.83  17.04  0.848  0.273  7.85  (1) (2) (3) (4) (5) (6) (7) (8)  T E M P E R A T U R E OF T E M P E R A T U R E CF TEMPERATURE CF T E M P E R A T U R E CF T E M P E R A T U R E OF CURRENT I I I N CURRENT 12 IN V C L T A G E V I IN  (8 )  INNER C Y L I N D E R AT P O I N T 1 0 . INNER C Y L I N D E R AT P O I N T 9 . INNER C Y L I N D E R AT P C I N T 8 . C I R C U L A T I N G O I L AT I N L E T POINT 1 1 . C I R C U L A T I N G O I L AT O U T L E T P O I N T 1 2 . AMPERES. AMPERES. VOLTS.  78  TABLE NO.  VOLTAGE BETWEEN THE  CYLINDERS=  9  6.92KV  '(I).  (2)  (3)  (4)  (5)  19.C5  1 9 . 05  19.06  16.70  17.02  Q.404  0.126  3.75  19.67  19.67  19.67  16.74  17.00  0.448  0.141  4.17  20.44  20.44  20.43  16.71  17.08  0.499  0.157  4.63  2 1 . 28  2 1 . 28  2 1 . 26  1 6 . 75  17.04  0.547  0 . 1 73  5.08  22.25  22 . 2 5  22 . 2 4  16.79  17.06  0.600  0 . 189  5 . 57  23.30  23.30  23.30  16.73  16.99  0.654  0.206  6.07  24.22  24.21  24.21  16.68  16.98  0.700  0.224  6.44  25.33  25.33  25.33  16.72  16.97  0.747  0.231  6.84  26.91  26.92  26.91  16.72  16.98  0.816  0.261  7.54  28.30  28.38  20 . 3 8  16.72  16\96  0.871  0 . 275  0 . 10  29.43  29.44  29.44  16.72  16.96  0.917  0.291  8.50  (1) (2) (3) (4) (5) (6) (7) (8)  T E M P E R A T U R E GF TEMPERATURE CF T E M P E R A T U R E OF T E M P E R A T U R E CF T E M P E R A T U R E CF CURRENT I I I N CURRENT 12 IN V O L T A G E VI IN  lb) '  il)  (8)  INNER C Y L I N D E R AT P O I N T 1 0 . INNER C Y L I N D E R AT P C I N T 9 . INNER C Y L I N D E R AT P C I N T 0 . C I R C U L A T I N G O I L AT I N L E T POINT 1 1 . C I R C U L A T I N G O I L AT OUTLET POINT 1 2 . AMPERES. AMPERES. VOLTS.  79  TABLE VOLTAGE  BETWEEN  THE  NO.  10  CYLINDERS=  7.91KV (8)  (2 )  (3 )  (4)  (5 )  (6)  (7)  1 9 . C4  1 9 . 04  19.04  16.66  16.92  0.402  0.128  3.72  19.75  19.75  19.75  1 6 . 67  1 6 . 89  0.452  C. 1 3 9  4.23  20.62  20.62  20.61  16.70  16.86  0.509  0.155  4.78  2 1 . 6T  2 1 . 68  2 1 . 67  1 6 . 68  16.90  0.560  0.168  5.32  22.83  22.82  22.82  16.67  16.92  0.627  0.196  5.83  24.22  24.23  24.23  16.71  16.95  0.700  0.224  6.43  25.57  25.57  25.57  1 6 . 74  1 6 . 94  0. 769  0.252  6.98  26.68  26.67  26.67  16.69  16.92  0.828  0.279  7.44  28.30  28.30  28.30  1 6 . 64  1 6 . 82  0.895  0 . 291  8.20  (I)  (1) (2) (3) (4) (5) (6) (7) (8)  TEMPERATURE TEMPERATURE TEMPERATURE TEMPERATURE TEMPERATURE CURRENT I I CURRENT 12 VOLTAGE VI  CF OF OF CF OF IN IN IN  INNER C Y L I N C E R AT PC I NT 1 0 . INNER C Y L I N D E R AT P O I N T 9 . INNER C Y L I N D E R AT P O I N T 8 . C I R C U L A T I N G O I L AT I N L E T POIMT 1 1 . C I R C U L A T I N G O I L AT CUTLET POINT 1 2 . AMPERES. AMPERES. VCLTS.  80  TABLE NO. 11 VOLTAGE BETWEEN  THE CYLINDERS*  9.63KV  (1 )  (Z )  (3 )  (4)  (5)  18.41  18.42  18. 43  16. 71  16.83  0.351  0.1.0P  3.25  19.05  19.06  19.06  16.69  16.89  0.4G2  0. 125  3. 73  19.74  19.75  19.74  16.65  16.89  0.453  0.144  4.20  2C.66  20.66  20.65  16. 59  16. 89  0.51.7  0. 166  4. 77  21.52  21.52  21.51  16.66  16.88  0.569  0.187  5.18  22.85  22.84  22.84  16.69  16.86  0.543  0.213  5.83  23.98  23.97  23.97  16.69  16.86  0.707  0.235  6.38  24.91  24.91  24.91  16.68  16.88  0.757  0.250  6.85  25.69  25.70  25.70  16. 67  16. 81  0. 8C0  0.264  7.25  26.68  26.68  26.68  16.65  16.85  0.855  0.279  7.32  27.53  27.52  27.52  16.63  16.81  0.898  0.293  8.24  28.72  28.72  28.73  16.71  17.01  0.953  C.300  8.84  29.65  29.65  29.66  16.71  16.85  0.998  0.312  9.94  11) (2) (3) (4) (5) (6) (7) (8)  (6)  (7)  (3)  TEMPERATURE OF INNER CYLINCER AT POINT 10. TEMPERATURE OF INNER CYLINCER AT PCINT 9. TEMPERATURE OF INNER CYLINDER AT POINT 8. TEMPERATURE OF CIRCULATING OIL AT INLET POINT 11. TEMPERATURE CF CIRCULATING OIL AT OUTLET POINT 12. CURRENT I I IN AMPERES. CURRENT 12 IN AMPERES. VOLTAGE VI IN VOLTS.  81  TABLE NO . 12 VOLTAGE BETWEEN THE CYLINDERS' 10.15KV (1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  17.74  17. 75  17.75  16.67  16.82  0.287  0.088  2.64  18.35  18.35  18.36  16. 66  16. 88  0. 341  18.9 2  13.92  10 .9 2  16 .69  16.82  0.391  19.70  19.71  19.71  16.68  16.85  0.449  0.138  4.21  20.55  20.56  20.55  16.69  16.83  0.5G7  0.161  4.70  21.21  21.20  21.20  16.67  16.86  0.550  0.171  5.02  22.12  22.13  22.12  16.70  16.88  0.611  C.198  5.47  23.01  23.01  23.00  16.65  16.83  0.661  0.215  5.93  23.83  23.83  23.82  16.63  16.00  0.700  0.231  6.35  24.62  24.62  24.62  16.63  16.89  0.744  0.246  6.77  25.57  25.57  25.57  16.66  16.79  0.800  0.264  7.25  26.43  26.42  26.42  16.65  16. 79  27.39  27.39  27.39  16.65  16.79  D.911  0.291  3.28  28.47  28.46  28.47  16.66  16.80  0.950  0.294  8.85  (1) (2) (3) (4) (5) (6) (7) (8)  0. 851  0. 103 0. 119  C.281  3.2 0 3.65  7.75  ' TEMPERATURE CF INNER CYLINCER AT PC I NT 10. TEMPERATURE OF INNER CYLINDER AT POINT 9. TEMPERATURE OF INNER CYLINDER AT POINT 8. TEMPERATURE CF CIRCULATING OIL AT INLET POINT 11. TEMPERATURE CF CIRCULATING 01L AT CUTLET POINT 12. CURREMT II IN AMPERES. CURRENT 12 IN AMPERES. VOLTAGE VI I N VOLTS.  82  TABLE  VOLTAGE  BETWEEN  THE  NO.  13  C Y L I N D E R S =, 1 0 . 80KV  (1 )  (2 )  (3 )  (4)  (5)  (6)  ( 7)  17.S4  17.95  1 7 . 95  16.66  16.76  0.310  0.095  2.88  18. A l  18.A2  18.42  16.61  16.77  0.354  0.108  3.30  1 9 . C9 " 1 9 . 1 0  19.09  16.59  16.77  0.408  0.126  3.80  19.77  19.77  19.77  16.66  16.78  0.462  0.147  4.27  20.45  2J.45  20.44  16.60  16.76  0.505  0.161  4.68  21.44  21.44  21.A3  16.61  16.77  0.568  0.181  5.20  22.14  22.1A  22.14  16.61  16.71  0.6C7  0.197  5.55  23.1C  23.10  23.10  16.61  16.76  0.662  0.216  6.05  23.83  23.83  23.02  1 6 . 57  1 6 . 68  0. 707.  C. 230  6.45  24.49  24.49  24.50  16.66  16.82  0.752  0.245  6.85  25.35  25.34  25.34  16.66  16.78  0.792  0.251  7.34  2 6 . 10  2 6 . 10  26.10  16.65  16.77  0.830  0 . 257  7 . 75  26.80  26.80  26.81  16.65  16.80  0.869  0.267  8.14  27.76  27.77  27.77  1 6 . 67  16. 76  0.919  0.283  8.63  28.72  28.72  28.72  16.62  16.81  0.975  0.298  9.06  (1) (2) (3) (4) (5) (6) . (7) (8)  TEMPERATURE TEMPERATURE TEMPERATURE TEMPERATURE TEMPERATURE CURRENT I I CURRENT 12 VCLTAGE VI  (8)  OF INNER C Y L I N D E R AT P O I N T 1 0 . CF INNER C Y L I N C E R AT P O I N T 9 . CF I N N E R C Y L I N D E R AT P C I N T 8 . OF C I R C U L A T I N G O I L AT I N L E T P O I N T 1 1 . CF C I R C U L AT ING O I L AT O U T L E T P O I N T 1 2 . IN A M P E R E S . IN A M P E R E S . ( IN V O L T S .  83  APPENDIX III  TABLES OE CORRECTED DATA  84  T A B L E N O 14  VOLTAGE BETWEEN  THE CYLINDERS= (3)  0.0  KV  (4)  (5)  (1)  (2)  16.35  16. 3 6  C. 0  0.0  17. 57  16.82  G. 2658  C.074 2  2.611  18 .01  16.98  0. 3078  0.0858  3. 03G  18.49  17.17  C. 3524  0.0900  3. 462  19.28  17.48  0. 4C48  0. 1100  4. COO  20.00  17. 75  c . 4487  0.1200  4.466  20.88  18.06  c . 49 86  0.1340  4. 9 52  21.90  18 .45  0. 5446  0.14C0  5.510  22 .97  13.95  c . 5966  0.1556  6.049  24.13  19.31  c . 6430  0.1638  6. 568  25.45  19.93  c . 69 74  0.1785  7.058  26.78  20. 29  c . 7482  C.1902  7. 584  27 .82  20.70  0.7860  0.2019  7.971  (1) (2) (3) (4) (5)  MEAN TEMPERATURE MEAN TEMPERATURE CORRECTED I I . CORRECTED 12. CORRECTED V I .  .  0.0  DF INNER CYLINDER. OF OUTER CYLINDER.  85  TABLE NO 15  VOLTAGE  BETWEEN THE CYL IN DEK S=  (4)  ( 1)  12)  17.88  16.94  0. 3C68  C.0898  2.947  18.42  17.15  c . 3561  0.1150  3.540  19.03  17. 42  0. 4C48  0.1260  3. 770  19.68  17 .64  0. 4527  0.1430  4. 199  20.4 7  17.98  c. 5044  0.1617  4.661  21 .23  18.29  c . 54 85  C. 1755  5.330  22. 17  13.67  c . 5947  0.1902  5.530  23.23  19. 11  c . 6440  C.2029  5. 990  24 .37  19.51  0. 6994  0.2206  6.520  2 5. 50  19.94  c . 749 3  0.2365  6.962  26 .70  20.32  c . 8C00  0.2553  7.415  27.82  20. 76  C.8432  0.2684  7.812  29.33  21.32  C. 8952  0.2864  0. 370  (1) (2) (3) (4) (5)  (3)  4.06 KV  MEAN TEMPERATURE MEAN TEMPERATURE CORRECTED I I . CCRRECT ED 12. CORRECTED V I .  (5)  OF INNER CYLINDER. OF OUTER CYLINDER.  86  TABLE  VOLTAGE  N G 16  BE T-WEEN THE CYLINOERS=  6.00 KV (4)  {2 )  19. 50  I 7. 69  C.4393  0.1410  4. 049  20 .45  18.05  0.4985  C.1578  4. 622  21.37  18.51  0.5534  0.1755  5. 160  22.35  18.84  0. 6024  0.1902  5. 610  23.26  19.23  0.5490  0.2049  6. 048  24.30  19.65  C.7003  0.2187  6. 559  25 .54  20.03  G.7622  0. 24 3 3  7. 048  26.78  20.44  C.7999  0.2533  7. 455  27.90  20. 82  C.8463  0.2733  7. 851  (1) <2) (3) (4) (5)  (3 )  (5)  (1)  NEAN TEMPERATURE OF INNER MEAN TEMPERATURE OF OUTER CORRECTED I I . CORRECT ED 12. CORRECTED V I .  CYLINDER. CYLINDER.  87  TAnLE  VOLTAGE  BETWEEN  THE  KO 17  C Y L I N D E RS = (3 )  6 . 9 2 KV (5)  (4)  ( 1 )  (2 )  19.05  17.48  C.4068  0 . 1280  3.780  19.67  17.74  0.4-506  0. 1430  4. 183  20.44  18.04  0.5014  0.1508  4.642  21.28  18.35  0.5484  0 . 1746  5.110  22 . 2 5  18.75  0 .6004  0 . 1902  5.600  2 3 . 30  1 9 . 08  G.6539  0 . 2069  6.097  24.21  19 . 3 6  0.6994  0. 2246  6. 461  25.33  19.81  0.7463  0 .2316  6.852  26.91  20.4G  C. 8 1 4 2  0 . 2613  7 . 544  28 . 3 8  21 . 0 7  0.8684  0 . 2753  8 . 100  29.44  21.49  0.9128  0 . 2 913  3.500  Q ) MEAN T E M P E R A T U R E OP INNER ( 2 ) MEAN T E M P E R A T U R E OF OUTER (3) CORRECTED II. ( 4 ) CORRECTED 1 2 . ( 5 ) CORRECTED VI .  CYLINDER. CYLINDER.  88  T A PL E NO 18 VOLTAGE  BETWEEN  THE  CYLINDER$=  7 . 9 1 KV (4)  (3)  (1)  (2)  1 9 . 04  1 7 . 43  C.4049  0.1300  3.747  19.7 5  17.69  C.4546  0. 1410  4. 247  20.62  10 . 0 3  0.5112  0.1568  4 . 788  21.67  18.43  C. 5 6 1 3  0.1697  5 . 350  22.82  10.90  0.6271  C.1971  5 . 860  24.23  19.41  C.6996  0.2246  6 . 452  25.57  19.88  C . 7 6 82  C.2523  6. 990  26.67  20.27  0 . 8 267  0.2793  7 . 445  28.30  20.95  C. 8922  0.2913  3 . 200  (1) (2) (3) (4) <5)  MEAN T.EMPERATURE OF INNER MEAN TEMPERATURE OF OUTER CORRECTED II. CORRECTED 1 2 . CCRRECTED V I .  {  CYLINOER. CYLINCER.  5)  89  T A C L E Nil 19  VOLTAGE BETWEEN  THE CYLINDERS= (3)  9.63 KV (4 )  (1)  (2)  18.42  17.20  C.3544  0.1100  3. 307  19.06  17.45  0.4C49  0.1270  3. 760  19.73  17.73  0.4557  0.1460  4. 390  20.66  18.02  C.5191  0.1677  4.777  21.52  18.37  0.57C1  0.1883  5.210  22.84  18.8 8  G .64 30  0.2137  5.860  23.97  19.27  C.7C6 3  0. 2356  6. 404  24 .91  19 .58  0.7562  0.2503  6.863  25.70  20.03  C.7990  0.2643  7.257  26.68  20.59  0.8530  C.2793  7.822  27. 52  21.00  C.8952  0.2933  8. 240  ^8.72  21.82  0.9493  C.3058  8. 840  29.65  22 .26  0.9980  0.3177  9. 235  (1) (2) (3) (4) (5)  (5)  MEAN TEMPERATURE OF INNER CYLINOER. MEAN TEMPERATURE OF OUTER CYLINDER. CORRECTED I I . CORRECTED 12. CORRECTED V I .  90  TABLE NO 2 0 VOLTAGE (1)  ;  BETWEEN THE (2)  CYLINDERS= (3)  1 0 . 1 5 KV (4)  15)  17.75  16.93  0.2906  C.Q8 8  2.682  1 8 . 35  17.17  C.3436  0.1050  3 . 246  18.92  17.39  0.3941  C.1210  3. 683  19.71  17.70  0.4517  0.1400  4.228  20.55  17.99  C. 5 0 9 3  C.1628  4.790  21 . 2 0  18.24  C.5514  C . l 724  5 . C5C  22.12  1 8. 60  0.6116  0.1190  5 . 500  23.01  18.88  C.6610  0.2157  5. 942  23.83  19.18  0.6994  0.2316  6. 373  24.62  1 9 . 57  C.7432  0.2464  6.784  25.57  19.99  C.7990  C.2643  7 . 257  26.42  20.48  0.8490  0.2813  7.752  27.39  21.02  C.9C83  0.2913  8. 283  28.47  21 . 6 7  0.9477  0.2943  8.850  11) (2) (3) (4) (5)  C  MEAN TEMPERATURE OF INNER MEAN TEMPERATURE OF OUTER CORRECTED II. CORRECTED 1 2 . CORRECTED V I .  CYLINDER. CYLINDER.  91  1ABLF: NO 21  VOLTAGE  BETWEEN THE CYLINDERS= (3 )  10.80 KV (4)  (5)  ( 1)  (2 )  17.95  16.97  C.3138  0.0968  2. 927  18.42  17.15  0.3574  0.1100  3. 340  19.09  17.40  C.41C9  0.1280  3. 82 8  19.77  17.67  0.4647  0.1490  4.287  20.45  17. 8 7  C.5073  0.1628  4.690  21.44  18.26  0.5692 : ; 0.1824  5. 230  22 .14  18.57  0.6074  '.' 0 .1980  5.580  23.10  18. 89  C.6618  0.2167  6.079  23.83  19. 18  0.7C64  C.23C6  6.472  24.49  1 9. 64  C.7512  0.2454  6. 863  25.34  20.08  0.7910  0.2513  7. 3 47  26.10  20.50  0.8284  0.2573  7.753  26.80  20.93  C. 8667  0.2 573  8.140  27.77  21.48  0.9158  C.2833  8.630  28.72  22. 04  C.9710  0.2983  9.058  (1) (2) 13) (4) (5)  MEAN TEMPERATURE OF INNER CYLINDER. F E AN TEMPERATURE OF OUTER \CYLINDER. CORRECTED I I . . CORRECTED 12. CORRECTED V I .  

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