IIZII A LABORATORY STUDY OF THERMAL CONVECTION UNDER A CENTRAL FORCE FIELD BY BHUVANESH CHANDRA B.Sc, University of Lucknow, 1961 M.Sc, Indian Institute of Technology, Kharagpur, 1965 M.Sc, University of Western Ontario, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of GEOPHYSICS We accept this thesis as.conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 19 71 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Geophysics The University of British Columbia Vancouver 8, Canada Date December 27, 1971. (i) ABSTRACT This thesis presents the results of a theoretical and experimental study of thermal convection under the influence of a central force field. Flows in the atmosphere and in the core of the earth are thought to occur under a near balance between Coriolis and buoyancy forces. Thus, a desirable model of these flows would include spherical symmetry in the force field and rotation. The present study, in which convection under a central force field in cylindrical geometry has been achieved, is the first step towards such a model. The system consists of a cool outer cylinder and a hot inner cylinder with a dielectric liquid (silicone oil) filling the annulus between them. The common axis of the cylinders is vertical. The inner cylinder is grounded and the outer cylinder is kept at a high alternating (60 Hz) potential. This intense alternating electric field provides the radial buoyancy force which results in convectiva heat transfer at a certain critical temperature gradient. The fluid in the system is found to behave like a layer of fluid in a gravitational field, heated from below. Below a certain critical value of a dimensionless number (equivalent to (ii) the Rayleigh number with the electrical force substituted for gravity) there is no convective heat transfer. Above the critical value, flow sets in with the convective heat transfer proportional to the modified Rayleigh number. Marginal stability analysis gives a critical electrical Rayleigh number in agreement with the experimentally determined value. (iii) TABLE OF CONTENTS ABSTRACT (i) LIST OF FIGURES (vLIST OF TABLES (vii) ACKNOWLEDGEMENTS (viiiCHAPTER I GENERAL INTRODUCTION 1.1 Introduction .1 1.2 Thermal convection and geophysical applications 4 1.3 Scope of this thesis. 7 CHAPTER II THEORY 2.1 Introduction 9 2.2 The perturbation, equations 11 2.3 -The principle of exchange of stabilities 23 2.4 The solution for the case of a narrow gap 27 2.5 The solution for a wide gap when the marginal state.is 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 Conclusions 67 REFERENCES 68 APPENDIX I TABLES OF THERMISTOR CALIBRATION 71 APPENDIX.II TABLES OF HEAT TRANSFER, MEASUREMENTS 74 APPENDIX III TABLES OF CORRECTED DATA 83 (v) LIST OF FIGURES FIGURE 1 Coordinate system and cylindrical cavity. 10 FIGURE 2 Dependence of the numerically determined electrical Rayleigh number on the dimension-less wave number. 32 FIGURE 3 . The velocity profile. 3 FIGURE 4 The temperature profile. 34 FIGURE 5 The stream lines at the onset of instability. 35 FIGURE 6 A cross section of the apparatus. 39 FIGURE 7 . The circuit for the heating coil. 41 FIGURE 8., Circuit.for the bridge. 44 FIGURE 9 Kinematic viscosity as a function of temperature. 45 FIGURE 10 ... Dielectric constant as. a function of temperature. 6 FIGURE 11 Dependence of the temperature of the outer cylinder less mean of inlet-outlet temperatures on heat transfer rate. 49 FIGURE 12 Calibration of Ammeter 1^ against a laboratory standard. 51 FIGURE 13 Calibration of ammeter I against a laboratory standard. 2 FIGURE 14 Calibration of voltmeter against a laboratory standard. 53 FIGURE 15 Dependence of temperature difference on heat ...... transfer rate for 0 kv rms between the cylinders. 55 FIGURE 16 Dependence of temperature difference on heat transfer rate for 4.06 kv rms between the cylinders. 56 (vi) FIGURE 17 Dependence of temperature difference Â°n heat transfer rate for 6.00 kv rms between the cylinders. FIGURE 18 Dependence of temperature difference, on heat transfer rate for 6.92 kv rms between tjhe cylinders. FIGURE 19 Dependence of temperature difference heat transfer rate for 7.91 kv rms between fche cylinders. FIGURE 20 Dependence of temperature difference on heat transfer rate for 9.63 kv rms between the cylinders. FIGURE 21 Dependence of temperature difference on hpat transfer rate for 10.15 kv rms between the cylinders. FIGURE 22 FIGURE 23 Dependence of temperature differej^pe on heat transfer rate for 10.80 kv rms between t^he cylinders. Nusselt number as a function of electrical Rayleigh number. (vii) LIST OF TABLES Table 1 Electrical body force at the inner and outer cyliners for given voltages across the gap. 22 Table 2 Dimensions of the cell. 4Table 3 Thermistor calibrations for the inner cylinder. 72 Table 4 Thermistor calibrations for the inlet and outlet points. 73 Table 5 Typical properties.of silicone oil. 48 Table 6 Observed data at 0 kv rms. 75 Table 7 . Observed data at 4.06 kv rms. 6 Table 8 Observed data at.6.00 kv rms. 77 Table 9 Observed data at 6.92 kv rms. 8 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. 1 Table 13. , Observed data at 10.80 kv rms. 82 Table 14. Corrected data for 0 kv rms. 4 Table 15 Corrected data for 4.06 kv rms. 85 Table 16 Corrected data for 6.00 kv rms. 6 Table,17,- Corrected data for 6.92 kv rms. 87 Table 18 Corrected data.for 7.91 kv rms. 8 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. 1 (viii) ACKNOWLEDGEMENTS The author is indebted to Dr. D. E. Smylie for directing the research reported in this thesis. The following persons read the thesis critically: Dr. D. E. Smylie, Dr. G. K. C. Clarke and DR. T. J. Ulrych, all of the Geophysics Department, and Dr. P. H. LeBlond of the Institute of Oceanography. Mr. H. Lau of the Mechanical Engineering Department contributed to many discussions. Finally, the author wishes to acknowledge the assistance of the staff members and the graduate students of the department who contributed in 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 last few decades, efforts have been made to study motions in the Earth's atmosphere, oceans, mantle and core through the use of laboratory models (Fultz, 1961; Long, 1954). The incentive for the use of .models comes from the difficulties 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 idealized conditions. Thus, laboratory models have become, important in geophysical fluid dynamics. The large scale circulation of the atmosphere occurs under the combined influence of a spherically symmetric gravitational field and the rotation ..of the Earth. .It is suspected-that a. roughly similar situation prevails. in the Earth!s liquid core.... Thus, a laboratory model of these flows.should .include both.the spherical symmetry of the gravitational field and the rotation of the Earth. The effect 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 in the atmosphere in the form of large scale eddies or cyclones. However, no .laboratory model which simulates a spherically symmetric 2 gravitational field has yet been developedo In the absence of a suitable method of simulating a gravitating fluid sphere in the laboratory, the validity of the rotating models is limited to the. polar regions where strong Coriolis forces are present as in the core and the atmosphereâ€ž In the thin shell of the atmosphere, however, this defect is not so serious because the flow is essentially two dimensional as opposed to the core flow which is three ,dimensional. This thesis is the outcome of a preliminary investigation for setting up a laboratory model of gravitational convection- Smylie(1966) first suggested the use of strong electric field gradients in dielectric liquids to simulate the central nature of the gravitational field. The electrical properties of a.liquid are functions of temperatureâ€ž For a nonpolar liquid the variation of dielectric, constant with temperature is due solely to the change in density (Debye, 1929, p 27)â€ž Thus, in the presence of temperature variations, in the dielectric liquid,.an intense electric field results in a buoyancy forceâ€ž This buoyancy force will produce an instability in the fluido However,.the viscosity, thermal,conductivity and the boundaries will act to stabilize the fluid and significant convective heat transfer should occur only when the temperature gradient is appreciable,, In dielectric liquids of moderate dielectric constant, the thermal convection produced by the electric buoyancy force is analogous to gravitational convection, where the instability occurs as a result of heating from below (or cooling from above). In contrast to the ubiquitous 3 and unalterable gravitational field, however, electric fields can easily be generated and shaped in the laboratory by existing techniqueso In a steady electric field space charges accumulate even in a good dielectric and the circulation of the liquid is due to the movement of these charges. The free charge buildup occurs exponentially in time with a time constant e/a where e is the permittivity and a is the conductivity of the fluid. This constant is known as the electrical relaxation time. If an alternating field is applied at a frequency much higher than the reciprocal of the relaxation time, free charge does.not have time to accumulate. Thus, in an alternating field, the dielectric effects dominate the free charge effectso The electric body force depends on the gradient of the square of the electric field* For most.dielectric liquids (for example, transformer oil and silicone oil), 60 Hz is a sufficiently high frequency to prevent the buildup of free charge. At the same time, dielectric loss at 60 Hz is so low that it makes no significant contribution to the temperature field. Further, variations in the body force are so rapid that its mean value can be assumed in determining fluid motions, except in the case of liquids of extremely.low viscosity. . Both theoretical and experimental study of electroconvection (convection under steady fields) has been reported by many authors (for example, Kronig and Schwarz, 1949; Turnbull, 1968; Malkus and Veronis,.1961; Melcher and Taylor, 1969; etc.).â€¢ The electroconvective 4 instability occurs even in the absence of temperature variations; Avsec and Luntz (1937) have observed two dimensional toroidal motions in a dielectric liquid filling the gap between two concentric cylinders. When the inner cylinder is earthed and the electrical potential of the outer cylinder is raised, then at some critical potential difference, steady cellular patterns are formed in the fluid. Recently, Gross(1967) and Gross and Porter(1966) have suggested the.use of space charge effects in modelling geophysical flows. In these models, charge transport becomes the analogue of heat transfer and. if .temperature gradients are present, heat will 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 stirs the fluid are not well under stood. Therefore, the quality of analogy to geophysical phenomena is not clear. 1.2 Thermal convection and Geophysical applications; Although the phenomena of thermal convection as a mode of heat transfer was discovered in the 18th century, the first quanti tative experiments on convection were done by Be"nard(1900). He found that if a thin layer of liquid, free at its upper surface, is heated uniformly at the lower surface, a regime of polygonal convection cells is formed as soon as a certain critical temperature gradient is reached...The walls of these cells are vertical and the movement of the liquid is upward in the centre and downward at the periphery. 5 Rayleigh(1916) laid the theoretical foundation for the explanation of Benard's results. It appears now, however, that the convection in Benard's experiments was actually partially driven by the variations in the free surface conditions and not entirely by the buoyancy forces as Rayleigh assumed. In later experiments (see for example Chandra, .1938; .Schmitd and Milverton, 1935; and Silveston, 1958), the free surface has been eliminated by placing an isothermal lid on top of the fluid layer. These experiments also showed the development of regular hexagonal cell patterns. The critical temperature gradient at which .convection cells develop i increases rapidly as the layer becomes thinner.. Several modifications of Rayleigh's theory (for example, Jeffreys, 1926; Pellew and Southwell, 1940; Low, 1929; etc.) to fit this case give excellent agreement with the experimental observations. Rayleigh's approach to the problem (infinitesimal, perturbations on an equilibrium state) still remains the basis of the modern treatment of the marginal stability problems (Chandrasekhar, 1961, chapter 2). It is believed that many flows of geophysical interest are driven by thermal convection. It has been suggested that large scale convection currents exist in the garth's mantle in attempts by many authors to interpret certain topographic features of the Earth's surface (Vening-Meinesz, 1962). Recently, McFadden(1969) has considered the effect of a region of low viscosity on thermal convection in the Earth's mantle to try to explain how convection 6 in only the upper mantle could lead to flows of continental horizontal scale. The presence of thermal convection in the atmosphere was suggested by Hadley (1735); differential heating between different latitudes gives rise to ascent of air in tropical regions. This leads to a flow of air towards the equator at lower levels and a flow away from the equator at higher levels. The combined effects of surface friction and the rotation of the earth deflects these currents and thus, according to Hadley's original theory, easterly winds form at lower levels and westerlies at higher levels. From this idea, the well known cellular model for the circulation of the whole atmosphere has been derived (Rossby, 1941). The geomagnetic field is now considered to originate from electric currents produced by inductive interaction between hydrodynamical motions in the Earth's conductive fluid core and a small adventitious magnetic field. Thus, a satisfactory theory for the Earth's magnetism requires a satisfactory theory for the hydrodynamics of the core. Several speculations have been made regarding the energy source that drives the fluid motions in the core. Among them thermal convection is one of the possibilities (Bullard and Gellman, 1954). Oceanographers have long studied the circulation of the oceanic currents. The ocean currents are believed to be the result of the combined effects of the thermohaline motions (density inhomo geneity caused by temperature and/or salinity differences) and the wind driven motions. The former are thought to be more important 7 in deep water. However, unlike the case of the atmosphere, the thermal convection in the oceans occurs, not because of heating from below, but because of cooling from the upper surface. The thermohaline circulation originates as a vertical flow sinking to mid-depth or even to the ocean bottom, followed by horizontal motion. 1.3 Scope of the present work This thesis is intended to demonstrate the feasibility of using an intense alternating electric field acting on a dielectric liquid to produce a spherically symmetric force field. A buoyancy force is produced by the variation of dielectric constant with temperature. Since it is a feasibility study, the experiment was set up in cylindrical.geometry to.simplify .construction. However, the techniques, learned here can be carried directly over to the spherical-case, which is the real geometry for large scale geophysical flows. . . The system consists of two vertical concentric cylinders of radii r^ and r^ ( r^ >rj_) â€¢ The .space between the cylinders is filled with a dielectric liquid (silicone oil). The inner cylinder is maintained at a fixed temperature T^ which is higher than the constant temperature T2 of the outer cylinder. In addition to a temperature gradient, a radial., alternating electric field is also imposed be tween . the cylin de rs. In chapter II, the marginal stability equations for this 8 problem are derived under the assumptions of the Boussinesq approximation. The effect of the Earth's gravity has been ignored for this theory. The solution of the marginal stability equations has been found numerically and it is shown that the convection induced by the electrical body.force occurs only when the dimensionless number (equivalent to the Rayleigh number with .the electric buoyancy force substituted for gravity), exceeds a.certain critical value. When the gap thickness is very much smaller than the mean radius of the cylinders (in this limiting case boundaries take the form of plane surfaces, rather than cylindrical surfaces),,the marginal stability equations are shown to reduce to the case of the Be"nard-Rayleigh instability. In.chapter III, the experimental arrangement, technique and results of electrically induced convection in a vertical annulus are described. The onset of thermal convection was detected by heat transfer and temperature measurements. The final experimental results, are shown,in the form of a graph of the Nusselt number against the electric Rayleigh number. The.study is summarized in the final chapter. 9 CHAPTER II THEORY 2.1 INTRODUCTION The problem to be studied is the thermal convection occuring under the electrical body force in the enclosed annular region formed by two vertical concentric cylinders of radii .r1 and (r^r^). This geometry is shown in figure 1. Conventional cylindrical coordinates (r,6,z) are indicated with z vertically upwards. The inner wall of the cylindrical cavity is held at a fixed temperature T.^ which is. greater than the constant temperature T^ of the outer wall. In.addition to. a temperature difference, there is also an alternating difference of electrical potential between the cylinders. In the absence.of any temperature variations, an electric body force is.produced in.the liquid,.and in an equilibrium situation, this body force is entirely, balanced by the generation of a pressure gradient (Landau and Lifshitz,, .1960, pp 64-69). The presence of temperature variations in the liquid results in buoyancy forces whereby warmer liquid has,a tendency to- seek regions of less intense electric field, cooler, liquid has a.tendency to.seek regions.of more intense electric field. However,-this natural tendency on the part of the fluid will be inhibited by its own viscosity. Thus, if the temperature difference FIGURE 1. CO-ORDINATE SYSTEM AND THE CAVITY. 11 between the vertical bounding walls is sufficiently small, the heat supplied at the inner wall is transferred to the outer wall by conduc tion alone. For greater temperature differences, the electric buoyancy force is sufficient to overcome viscous dissipation and the fluid assumes a convective motion. In other words, we expect that the temperature difference between the cylinders must exceed a certain minimum value before convective heat transfer is realized. The problem under consideration, then, is to solve the marginal stability equations for an incompressible fluid layer bounded by vertical cylindrical walls at different temperatures, in the presence of a radial electrical buoyancy force. 2.2 The Perturbation Equations The following simplifying assumptions are made in deriving the perturbation equations: (1) The fluid is incompressible and the density constant, except as it modifies the electric-buoyancy force term. (2) The mean square electric field and the temperature distribution in the annulus are functions of radius only. (3) The cylinders are of infinite extent, so that the end effects can be ignored. (4) The motion is slow and the components of velocity are small enough so that in a first approximation, their products and squares can be neglected. 12 The kinematic viscosity and the ,.dif fusivity of the fluid are true constants. The various perturbations are axi^symmetric and thus independent of 0. In deriving the. marginal stability .equations, the effect of the earth's gravitational field can be ignored. Strictly speaking it is .not so.. Batchelor. (19.54). showed, that-in .a.narrow cavity between vertical, boundaries at different ..temperatures, there is a- slow vertical flow but the heat transfer across the cavity is due mostly to conduction. Thus, the present problem is unstable for all, 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 is roughly 3 .10, .times, less than the ..Grashof. number....at which free convection sets -in.. (Vest, and Arpaci, . 1969)...... The gravity induced, flow is likely to be very small., at. the.. Grashof ..number involved in the experiment. It would, be an important generalization .to carry through a solution taking into account the interaction of the convection, induced by the electrical forces with the vertical gravitational base flow,,, but this difficult problem has not been tackled here. Consider that initially there are no motions. Thus, the initial 13 state is characterized by U = 0 T = T(r) (1) where U is the velocity vector and I is the temperature. In'the absence of any .motions, the hydrostatic equation reads -grad p + 1 = 0 (2) o o -Â»â€¢ where f -is . the electric body, force per unit volume at density p and temperature T : p is the pressure distribution, o o o r , . The .temperature distribution...is governed by the equation . <Â» The solution of equation (3) appropriate to the boundary conditions gives T T T dTQ 1 2 1 dr râ€ž r (4) in/i. "1 The solution for the electric field distribution E is o Ej = (Eq,0,0), (5) . o r2 r (6) where Vq is the root mean square value of the applied potential difference. 14 Let the perturbed state be characterized by the velocity components V ue, V (7) body force f = + t\ (8density p = pQ + p , (9) electric field E = E + E , (10) o i temperature T = TQ + T , (11) i and pressure p = PQ + P , primes indicating flow induced quantities. The. general equation for the body force resulting from electric fields in fluid dielectrics .is developed by Landau and Lifshitz (1960,,p.68) - They showed that in an uncharged fluid dielectric of uniform .composition,.,the .electric body force, per unit volume is given by V= 1/2po &rad[Eo (-If }T ] " 1/2Ed ( If )p ^ad To , (12) o o where e is the electric permittivity of the fluid. In the perturbed state at temperature T and density p, the body force.per unit Aolume is 1 = 1/2 pgrad [E2 ( -g- \^ - 1/2 E2 ( || )pgrad T (13) 15 The permittivity of a substance is a function of density and temperature. If the perturbations are small, the permittivity at temperature T and density p can be expressed in terms of permittivity at the reference temperature Tq and density pQ by writing a Taylor expansion. Thus, +1/21 $ v'2+2 c \-*.p,r + c d \T'2 ] 3p o oo 3i o + higher order terms " (14) Therefore, we can write , 3e' ( 3Â£ . . d2c . , . 32e . , ^ 3p ;T S 1 3p ;T - 2 T P ^ 8p8T ;p ,T o 3p . o 00 and 31 p 31 po 3T2 Po 3p3T po'To Combining (8), (12), (13) and (15), we get to first order in the flow-induced quantities, f* - l/2p'grad[Eo2 ( ) ] - 1/2EQ2 (|| ) gradT' o o + pograd[(2o.^) ( fÂ£- )T ] - (E;.I') ( || > gradTo o o .+ l/2pograd[Eo2 {(^f )T p' + ( ) T'} ] 3p o o o -1/2E/I (4 >p *' + (fe)0 Â»']Â«Â»Â«. <Â») 3T o o o 16 Chandra (1969) showed that for a fluid dielectric of moderate dielectric constant, the flow-induced changes in the electric field can be neglected. The order of magnitude of the second density and temperature coefficients of permittivity for fluids is very small and can also be taken to be zero. . Thusthe. electric buoyancy force per unit volume can be written as . f - 1/2 p' (f^->T gradEQ2--iÂ°- ( || )p grad' T' (17) with p' - - ctpT'5 (18a denoting the coefficient of thermal expansion. The equation (17) is quite general and can be applied to a fluid dielectric .of any shape.. . Using equation .(6), the electric buoyancy force per unit mass for the cylindrical case under consider ation can be written as v 2 r2 Sp 'T rJ Â» r [ln-^] rl (19) where r is the unit radius vector. Having found.an expression for the electric buoyancy force per. unit mass, the linearized perturbation.equations:can be written as follows: (i) momentum 17 at"" v[v ue "-I 1 <21) r 2 Ci TT V z- 9 / Â£l x 4. o2TT i /. o 1 , 3e . 1 3T* Â° (In â€”r r rl (ii) continuity 3U U 3U dr r dz (iii) energy TrT2 \ _ 3t r r ln â€” rl KV*T' (24) 2 where V has the meaning 2 2 V2 = -â€” + - â€” + -â€” (25) a 2 + r 3r + a 2K J 3r 3z and v and K are the kinematic viscosity and the thermal diffusivity of the fluid. By analysing the disturbance into normal modes, the solutions of equations (20)-(24) are to be found in the form Ur = eptU(r) coskz, T1 = ept0(r) coskz, U = eptV(r) coskz, p* = eptfi(r) coskz, V(26) Po U = eptW(r) coskz. z where k is the wave number of the disturbance in the axial direction 18 and p Is a constant which may be complex. For solutions of the form (26), equations (20)-(25) reduce to V 2 v(DD,-k2-P/v)U + â€”2-- [a( f ), % - -f- ( jÂ§ ) \ jf ] - DO, (27) (In-2-)2 rl (DDA - k2 - p/v)V = 0, (28) V 2 v(D*D-k2-p/v)W + -fr Â°â€” ( || ) *f = -kfl, (29) 2p r 3T p r2 (In â€”) rl D^U = -kW, (30T -T 2 , vâ€ž 1 2 U (D^D-k"-p/K)0 - - -^-f ^ , (31) i 2 icln â€” rl and V2 = D^D-k2, (32where D = â€”- (33) dr and D* - h+ r' (34For the case when p=0 the solution of equation (28) is V = AIQ(kr) + BKQ(kr) (35) where I and K are modified Bessel functions of zeroth order, o o Substituting the boundary conditions that on rigid boundaries V=0, we get A = B = 0. Hence, V = 0. (36) This shows that for the form of assumed solutions, there cannot 19 be any velocity perturbation along 9, Thus, the basic flow is in the form of ring vortices. Eliminating W between equations (29) and (30), we get -| (DsI)-k2-p/v)D,,U + -^T C|f)p if.-kfi (37) Substituting the above expression for o, in equation (27), we get after some.simplifying steps V 2 2 (DD,-k2-p/v)(DD,-k2)U- -2- Io(.|Â£ >T - Â± ( || > ] % (ln -*x2 r For fluids of uniform composition, the permittivity 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 (DD^-p/vXDD.-k2)^ - - ~\ Â£ < || ) ^ (40) (In/)2 1 Equation (40) must be solved together with T -T (D.D-k2-p/<)0 = - â€”Â±â€”r '7 (41) r2 â€¢ <ln â€” rl Equations (40) and (41) are the marginal stability equations, the solution of which must be found under the boundary conditions â€¢ U =..DU = 9 Â» 0 (42) at pr^ and r8*^. 20 Dimensionless parameters to describe solutions in general form may be obtained by non-dimensionalizing the equations,in a suitable manner. A convenient set of non-dimensional variables is r - r. *\ J2Â£ > (43) a =Â» kd A Â» Â£ where the cavity gap - is written as d. The equations (40) through (42) become (DD#-a2-o)(DD#-a2)U 2 2 a d __1 , 3e > e Ilnf2]2 rx3 ^ 9T P (1+AO3 (44) (D.D-a -Pc)G -â€¢-TrT2 d2 â€¢cin r2 rx (1+X?) 'â€¢ (45) U = DU â€¢Â» 9 0 at C B 0 and 1. (46) D and now have the following meaning de: â€¢ (47) (48) and P (Â» â€” ) Is the Prandtl number* It is convenient to make the transformation KT r Inâ€” 9 +6, (Tj-T^d2 rl where 0 now has the dimensions of velocity. The marginal stability equations become (DD.-a2-o)(DD.-a2)U= -Ra2 9 (1+U) and (D*D-a2-Pa)0 = ^ R may be interpreted as the electrical Rayleigh number ag Bd4 R = VK where y 2 o 1 , je_ . *e " " 3n r2,2 p a 8T P ^(Inâ€”) can be regarded.as an electrically derived .gravity at the surface the inner cylinder and T -T 1 2 6.- r2 rlln ^ is the temperature gradient at the inner cylinder surface. Clearly, the electrical gravity, g^, is a strong function of radius and varies inversely as the cube of.radius. 22 Table 1 The electrical body force at the inner and outer cylinders for a given voltage across the gap. r, - 1.711 x 10_2m . .e =8.854 x 10~12 farads/m 1 o r2 â€¢ 1.903 x 10~2m p = 937.7 kg/m3 _3 a â€¢ 1.08 x 10 cc/cc/Â°c 3 , e . â€ž -,0 -,n~3/o ( â€” ) = 3.72 x 10 / c o VOLTAGE ACROSS Electrical gravity... . Electrical gravity CYLINDERS ... Earth's gravity ' y Earth's gravity kv(rms) at r=r^ at r=r 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 electrical 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 electric gravity is three times that of the earth's gravity. Above this voltage difference, it rises even more sharply. .The form of the equations. (50) and (51) and the boundary conditions (46) shows that, in the marginal state (i.e. when a=0) , for a given value 23 of a, the dimensionless parameters whose values are sufficient to determine uniquely the distribution of U and G are the electrical Rayleigh number R and the gap to inner radius ratio X. 2.3 The Exchange .of Stabilities In general o" is a complex quantity and is a function of the physical quantities involved and of the parameters characterizing the particular pattern, of the disturbance. If Rl(a) is positive, the disturbance increases, but if it. is .negative, the disturbance dies away. In a.set of possible disturbances, the mode for which Rl(o)is. a-maximum will, be the first to appear beyond stability, when the Rl(o) is just equal to zero, the limiting condition of stability will.be realized. It.is important to know.if the Im(a) is zero when the Fl(a)is. v If .Im(a) is not zero, the disturbance manifests itself .in the form. of. a wave.motion. and the system is said to be overstable. -When Im(o) is zero if the Rl(a) is zero, we say that the principle of exchange of.stabilities is valid. Since the vanishing of .cr means that all time variations disappear, this limiting condition represents a steady state in which the disturbance just maintains itself. To consider, the principle of exchange of stabilities for the problem in hand,.it is.slightly .more convenient to write the non-dimensionalized form of equations (40)-(42) with respect to 24 r^, the radius of the outer cylinder. Then, we have (DDA-a12-o1)(DD^-a12)U = ^3 (55) r <D*D-al2-P0l)Q = 7 (56and U = DU = 6 Â» 0 at r = n and 1. (57) where D - ^ , D* - ^ + Â± , (58ar al = kr2 " -T â€¢ (59) a R, " ~r and n - TT1 . (61) XH 2 Multiplying equation (55) by rU* (the star superscripts represent complex conjugates) and integrating in the interval r = n and r =1, we get f rU*(DDA-a12-a1)(DDA-a12)U dr = -R^ [ ^ U* dr (62) in r Since U and its derivative vanish at the boundaries, the left hand side of equation (62) can be shown to be positive definite (Chandrasekhar, 1961, p297). Thus j rU*(DDA-a12-a1)(DD^-a12)U dr = f1 r|(DD^-ai2)u|2dr + orf[r|g|2 + (^ + a^r)!^2] dr (63) 'n Jn 25 Substituting for U* from equation (56.) in the right hand side of equation (62) , we get Â£ ^ U*dr = ln f (n*D-a12-Pcv*)0*dr (64) = f1 - DJ)0*dr - (a 2+Po*) t L dr (65) Jn r * 1 In r Again making use of the boundary conditions on 0 and integrating by parts, we.can easily show that J1 | D^D0*dr = - ^ |D0|2dr + %:> D9*dr (66) Combining equations (62), (63), (65) and (66), we get + I2 = R1a12t(a12+Po*)I3 + 1^ + Ig] (67) where Ix = J1 [r|-BU|2 + ( i+ a2r)|u|2] dr (68) I2 j^.. r|(DD^-a2)u|2dr (69) 26 12~t) ^lefdr (70) [â€ž i|D6|2dr (71) and I5 = -2 f1 0_ D9*dr (72) Jn 2 1 r The integrals 1^, , I^ and 1^ are positive definite while !<. is complex. Equating the imaginary parts of equation (6 7), we obtain Im(a) [I1 + Ria;L2PI3] = R1a12Im(I5) (73) and no general conclusions regarding the stability of flow can be drawn from this equation. The validity of the exchange of stabilities cannot be rigorously;proved for this problem. In .the next.section, it is shown that, for a narrow gap, the stability equations reduce to the form of the. Rayleigh-Benard case.. The principle of the exchange of stabilities for the Rayleigh-Benard form of equations has been proved elsewhere (Chandrasekhar, chapter II).. .In.chapter III, the experiments on electric body force, induced convection are described and they show that the instability appears to set in as a stationery coavactiva flew. Without attempting a.theoretical justification of the principle of exchange of stabilities, we shall assume its validity 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 r0-r (~d) between the cylinders is small compared 2, 1 r "tÂ°r to their mean radius 1 2(=r ) ,D. can be simply replaced by D, â€”2â€” o * (Chandrasekhar,1961, p 402). In this scheme of approximation, equations (40) and (41), become â€ž 2 9 2 9 9 V (D -a -a)(D -aZ)U = - \ Â° ad ( 3e . Â©_ (74) r~~ "vp â€¢ 1 8T ;p r 3 [ln â€”j2 o V 2 and (D2-a2-Pa)G =- <Ti~T2)d JJ ' (75) râ€ž r , 2. O Kin â€” â€¢ rl where 5 - H!i . k = f , P/v = ^ (76) d d The boundary conditions (42) are U = DU =. 0 = 0 at Â£=0 and 1 (77) By making the transformation (T-T.)d2 - â€¢ 1 â€¢ U+ U (78) icr In â€” Â° rl equations (74) and (75) become (D2-a2-o)(D2-a2)U = -R0 (79) (D2-a2-Po)0 - U (80where , . . K- â€” (81) VK 28 gg and g are now defined at the mean radius r^ and have the following meaning ^ K---r-^-21 ^<H>, <82) T -T and g = 1__2_ (g3) i r2 r In â€” rl Equations (79) and (80) are the same as for Rayleigh-Benard instability (Chandrasekhar, chapter 2)â€ž The principle of exchange of stabilities for this case holds and the instability occurs at a Rayleigh number 1707., 762 corresponding to a = 3oll7â€ž This analysis serves to show that the principle of exchange of stabilities is.valid in the small gap approximation. However, it ignores the curvature effect of the cylinders and cannot be used for any comparision with the experimental results. 2o5 The solution for a wide gap when the, marginal state is stationary, In the marginal state (a=0), the equations (55) and (56) become _ â€ž on (DD* - a^ru = -R.^^ (84r and (D*D - aÂ±2)0 = ^ (85) On eliminating U between equations.(84) and (85), we obtain QVI +. 7 QV + 5_ _ 3 2 IV _ ^ + ]W_ III r 2 1 3 r r r .2 _ 2 4 + ( -j - â€” + 3a ) 0 - ( -j - â€” â€” ) 0 r r r r - a.66 - -R^2 ^- (86) 1 114 r 29 with the appropriate boundary conditions 2 II J I 2 III .2 II 2 1 al ft 0 = 0 - &1Q = Q + -p - ax 0 -0 = 0 (87) rl at r 1 â€” and 10 r2 Alternately, we can eliminate 0 between equations (84) and (85). The eigenvalue problem, then., becomes 2 2 TTVI -4-1^.4. I 16 *o 2>UTIV Â± f2 ^l W.IH ,3 , 23S1 - 4...II U + -U + (â€”-pp â€¢Â» 3a )U + (^ â€”)U -(â€” + râ€”- 3a. )U r r r r . + !^.)Di _(3_ _ â€ž. Z!i_ + 6)0. 2 u_ (88) r r r r r r subject to boundary conditions U.= U1 , UIV + |um - (\ + 2a12)U11 + (^ - ^U1 -(\ - ^ -a^U 5 0 r r r r at r - â€” and 1. r2 The critical values of and a^ were found by solving numerically equation (86) with boundary conditions (87). The. general solution of ..equation (86) is 0 = A1Â©1 + A202 + A3Â©3 + A404 + A$Â©5 +A606, where A.^.. .Ag are constants. The condition that these solutions 0 ,0,.,,8, form a 1 z o fundamental set is that their Wronskian is not zero. There is an (89) 30 infinite number of possible sets of solutions. One simple set pai* be found by choosing 0^(r) such that 01(rI/r2) = 1, Â©.^(^/r,,) = Â©^Vi/O = ....-6^ {T^TJ = 0 (90) and defining 0 (r) where n = 2,3.. <6 as that solution which satisfies n the initial conditions 0 ^(r) Â« 0 If m 4 n, ra â€¢-â€¢ 1,2...6 (91) n 0 m~1(r) = 1 if m = m m = 1,2...6 (92) n Then 0,(r), 0_(r).... 0, (r) form a fundamental set and the value 1 2 6 of their Wronskiah at r = â€” is unity (Ihce,. 1956jp 119). ! r2 For a. given, value of a^ and. R^. the solutions 0^ .(Â£),, 0^(r) ... 0^(r) , with the initial conditions (90)-(92), were- found numerically bf Xhe method, of Runge-Kutta by breaking the sixth .order differential equation (86) into six simultaneous first Order equations' (Ford, 1955, chapter 6) by writing <z>'1 =â€¢ G 011 - H em = J 0IV - K 0* = L and L1 + ^- L + (^ - 3a 2)K - (^ + ^ * )J + (^ - + 3a^)U 2 r 4 r r r o 5a1 7a 2 -(^5 - -3 "-)G + (R/i_ Â¥ a.6)0 = 0 (93) r r r 4 ' 1 r 31 Knowing 0,(.r), Qn(r) ... 0, (r) and thus 0, the boundary 1 2 o conditions (87) were substituted to give a set of six linear homogeneous equations (three at the inner boundary and three at the outer boundary)., The condition that, they have a non-trivial solution is that the determinant DET= all a12 a, a13 a14 a15 a16 21 l31 41 51 a61 a62 a63 %6 - f(a1,R1) i 0 (94) where a,, (j; = I...6) aire the coefficients- of A. in the first 1.1 J' 3 equation; a^j are coefficients of:in .the..second,.equation and so on. For a given.value pf-a^, some .value of will cause the determinant to vanish. .To find that value, determinant (94) was calculated for .increasing values pf R^ till the determinant changed sign..... Knowing ..positive^ ..and- ,.negat;iy& -values, 'of the determinant, corresponding to two values pf %.^*. & linear interpolation scheme (regula falsi, method, Mathews and Walker, 1964, p339) was used 3.08 3.10 3.12 3.14 DIMENSIONLESS WAVE NUMBER a FIGURE 2. DEPENDENCE OF THE NUMERICALLY DETERMINED ELECTRICAL RAYLEIGH NUMBER ON Ti i'L DI KENS! G,'<LL".r>:. WAVE NUMBER. 33 FIGURE 3- THE VELOCITY PROFILE AT THE ONSET OF CONVECTION. THE VELOCITY IS NORMALIZED TO UNIT MAXIMUM AMPLITUDE. 34 FIGURE 4. THE TEMPERATURE PROFILE AT THE ONSET OF CONVECTION. THE TEMPERATURE IS NORMALIZED TO UNIT MAXIMUM AMPLITUDE. 0.0 0.2 0.4 0.6 o.e 1.0 FIGURE 5. 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. 36 to iterate and find for which the determinant vanished. R^, in general, converged to a finite value (to eight significant figures) within six iterations. 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 in figure 2. The solution of interest is the minimum value of R for which f(a,R) = 0. Quadratic iteration was used here to find the value of a for which R is minimum. The method has been described by McFadden(1969). The critical values of R^ and found numerically are R = 2119.346 ;. a = 3.117 c c using the values r^ = 1.711 cms.; = 1.903 cms. for the radii of the cylinders. These values were also calculated by solving the sixth-order equation in U and were found to agree within three decimal places (seven significant figures). Figure 3 shows the profile of the velocity distribution corresponding to L and a^. Figure 4 shows the temperature distribution. The stream lines at the onset of instability are shown in figure 5. 37 CHAPTER III EXPERIMENTAL ARRANGEMENT AND RESULTS 3.1 Introduction Chandra(1969) has described the cylindrical experimental arrangement in detail. He also presented some preliminary results oh electrical body force driven convection. Although the heat transfer measurements clearly indicated the change from conduction to convection regime, the measured Nusselt number (ratio of total heat transfer to heat transfer by conduction alone) In the conduction regime was found to be 10-15% less than its value of unity. No definite explanation for this discrepancy could be given. However, it was suspected that the discrepancy arose partly due to the heat losses from the ends of the heating coil and partly due to the errors in the inference of the temperature of the outer cylinder. The outer cylinder was cooled by transformer oil kept at a constant temperature by circulating it through an external thermostatic bath. The temperatures at the inlet and outlet points (figure 6, 11 and 12) to the thermal flask were measured and their mean was taken to be the temperature of the outer cylinder. However, for a short contact time of flowing liquid at a temperature lower than that of the solid wall, the fluid temperature changes appreciably only in the immediate Vicinity of the wall (Bird, Stewart and Lightfoot, 1966, p349). Away 38 from the. wall, there is hardly any change in fluid temperature even when there is increased heat transfer between the cylinders. It was therefore, felt necessary to do the experiments again to be able to compare experimental results with theoretical values. In.these experiments, the heating coil was placed much closer to the wall of the inner cylinder to minimize heat loss from the ends and to assure radial heat transfer. However, heat transfer and temperature measurements showed that, this could not account for the observed discrepancy. Next, temperature of the outer cylinder was measured at zero applied potential independent of. the inlet and outlet temperatures. The true temperature of the outer cylinder was found to be different from the. mean. A.linear relation was obtained between the mean of the inlet and outlet temperatures and the true temperature of the outer cylinder- This was done only when there was no potential difference across the cavity. When the outer cylinder was at a high voltage,.Its temperature could not be.measured directly. However, from the linear relationship, the temperature of the outer cylinder was found indirectly under the assumption that the flow and temperature distribution in the thermal flask are not altered by the high voltage. The Nusselt number now gave a value of unity in 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 finish to avoid any sharp points which might cause electric breakdown at high voltages. The inner cylinder (2) was,constructed in two parts. The central part was made.from brass tubing in which holes at 8, 9 and 10 were drilled to accept thermistors,, A tight fitting stainless steel tubing was then forced over the brass tubing. End pieces were machined from stainless steel and fitted to the central part with inside corners at the lines of contact to prevent the occurrence of corona. The heating coil.(7), made of a close wound 30 S.W.G. diamel coated'Mancoloy 10' wire on afibreglass rod, was a close.fit inside the inner cylinder. After installing calibrated thermistors in the receptacles at.8, 9.and 10, the. heating coil was centered, and the remaining space inside the inner cylinder was filled with thermal epoxy and allowed to set. Besides the end points, the heating coil was also tapped at two intermediate points directly opposite the thermistors at 8 and 10. The leads for the thermistors and the heating coil 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 measure ment length between the points 8 and 10. The outer cylinder was. equipped with toroidal terminations to prevent the occurrence of â€”G> POWER SUPPLY R2 VW 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 Â± 0.002 cms. Outer radius = 1.711 Â± 0.002 cms. Radius of the .outer cylinder Inner radius = 1.903 + 0.002 cms. Outer radius = 2.114 Â± 0..002 cms. Length of the measurement section between the points 8 and 10 L = 8.25 Â± 0.02 cms. The assembled apparatus was set in a double walled thermal flask (5) filled with transformer oil. This oil was circulated through an external thermostat, thus enabling the temperature of the outer cylinder to be maintained constant. The temperature in the thermal flask could be controlled to Â±0.02Â°C. The present.setup didnot allow to observe fluid motions visually. The onset of thermal convection was detected by making heat transfer and temperature measurements. The Heating Circuit: The control circuit for the heating coil is shown in figure 7. As already mentioned, the heater was a three sectioned heating element. Currents in the three sections were adjusted by means of the rheostats R^ and until the thermistors at 8, 9 and 10 gave the same.temperature reading (control to about 0.01Â°C was found feasible) to ensure uniformity of.temperature 43 on the surface .of .the inner cylinder. At equilibrium the heat transfer occurring in the..section.between the points 8 and 10 was taken to be equal to the dissipation of electrical energy in the central section of the coil. The ammeters 1^ and and the voltmeter were all of .1% accuracy (made by Weston, Model 81). The current for the heating element was drawn from a direct current, (Hewlett Packard, model 6290A) power supply. The power input in the central, section .of the coil.is. given by (i-j.. - i2) v1 watts The Thermistors: The glass coated thermistors of .043 in. diameter had a nominal resistance of 50,000 Â± 20% ohms at 25Â°C and a dissipation.constant.in air of 0.7 milliwatts per Â°C. All thermistors were calibrated before use. The calibration was per formed against a.N..P.L..calibrated platinum resistance thermometer. The.calibration points were recorded over a 15-28Â°C interval. The results were analyzed,by the. method .of. least squares for fit .to. the. orthogonal, polynomials for the function lnR =.A + B/T + C/T2 where ,R .is... the. resistance of the thermistor at temperature TÂ°C and A,- B...and. C. are .the. constants, of .the .cubic equation. .The results of calibration and the difference between the calibration, resistance and. the...computed resistance. are listed in tables 3 and 4 (Appendix 1). -o- -O o Sj = 29992.3 OHMS S2 = 29992.8 OHMS R DECADE RESISTOR T, ,T2 ETC. THERMISTORS FIGURE 8. THE BRIDGE CIRCUIT. 45 â€¢ 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 direct reading bridge specially designed for this purpose (figure 8). The High Voltage Supply: In all 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. A voltage stabilizer was inserted between the mains and the input to the transformer. This eliminated the fluctuations of the power line and gave a constant output voltage. A high accuracy (1%) electrostatic voltmeter was used to measure the applied potential difference between the cylinders, The Dielectric Liquid: The dielectric liquid for these experiments was DC200 electrical grade silicone oil (supplied by Dow Corning) whose physical.properties are listed in table 5. The silicone oil was selected because it has a high electric strength and is available commercially in any desired viscosity grade. The dynamic viscosity and the dielectric constant of the oil as a function of temperature were measured using a falling ball viscometer and a standard capacitive cell respectively. In figure 9, the kinematic viscosity of the oil is plotted as a function of temperature. Figure 10 shows the variation of dielectric constant with temperature. Other properties listed in.table 5 are taken from Dow Corning Bulletin 05-213 of July 1968. 48 Table 5 Physical Properties of Silicone Oil* Density,. p--0.9377 gm. cm -3 Coefficient: of volume expansion, a- 1.08x10 Thermal conductivity, x = 3.2x10 -4 cal. cm -1 sec Specific heat,. Cp= 0.43 cal. gm. Dielectric constant, K =2.64 e Kinematic viscosity,.v = 12.225 cm sec. -1 *A11 properties are given at 23Â°C. 3.3 Experimental Procedure : The cylindrical annulus, previously cleaned, was filled with degassed electric grade silicone oil. The cavity was.further connected to a vacuum pump to remove any tiny bubbles sticking to the walls of the cylinders. Having made all electrical connections, the external thermostat was switched on. . In all experiments, the temperature, of the outer cylinder, T^, was kept lower than the temperature of the inner cylinder, T Â» cylinder, T^> with the mean of the inlet and outlet temperatures, was done without any high voltage across the cylinders. For this, 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 in the three sections of the heating coil were adjusted An initial run, to correlate the temperature of the outer -n <r> 70 rn Iâ€”1 2 tn z o o TI ro â€” o z m rn H i o c â€”i r~ rn O â€”I u rn m â€”i -6 m m -i 5 rn H <? c > 70 â€”I rn X m > LO rn 70 70 > â€”I m o -Tl m o c â€”I rn 70 o -< z o m 70 r~ m tn tn TEMPERATURE CORRECTION IN CENTIGRADE DEGREES 6<7 50 with rheostats R^ and (figure 7) until all the temperature readings on the surface of the inner cylinder were within 0â€ž01Â°G. The temperature of the outer cylinder and that of the inlet and outlet points,were all noted. After 10 minutes, all temperatures were measured again and if there was.no change then the readings were accepted. The procedure was repeated for several values of heat transfer.rate across the. cavity.. It was found that the actual temperature of the outer cylinder less the mean temperature read by the inlet and outlet thermistors (11 and ,12 in. figure 6) is linear in the heat transfer.rate between the. cylinders as shown in figure 11. . , When the. heat transfer, and.temperature measurements were done with the outer cylinder, at.a.high alternating potential, it was not possible to .measure its.temperature. Since the flow and heat transfer1 .in .the circulated..transformer oil filling the thermal flask . .iare. likely, to.be,'largely, unaffected, by the ..potential of the outer cylinder, :itwasi;assumed that .this, is precisely so. Knowing the . heat transf er..,rate..and. the mean, of ..inlet-outlet temperatures, the plot, shown -in, figure. 11 provided, an indirect means of obtaining the temperature of, .the, outer cylinder. ...The experiments.were run for.the. following values of potential 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 MA 100 MA RANGE FIGURE 12. DIFFERENCE BETWEEN THE READING OF AMMETER ^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. Y IN MA +10--10 X IN MA 100 MA RANGE X IN MA 300 300 MA RANGE 0.6 X IN AMP 1 A RANGE FIGURE 13. DIFFERENCE BETWEEN THE READING OF AMMETER I2 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. Y IN VOLTS +.05 --05 X 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 All the observed data is tabulated in tables 6-13 (Appendix II). 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. The results of calibration are shown in figures 12-14. All values of currents and voltages were reduced using these graphs. The corrected values of i , i and v and the average temperatures of the inner and outer cylinders are tabulated in tables. 14 through 21 (Appendix III). :Errors-,..due..-to. self heating of ..the, thermistors: The thermistors were specified to have a. dissipation, constant in air, of..0.7 milliwatt per Â°C. Accordingly, a, .power, of .0.7 .milliwatt in..the thermistor will change its- temperature by. 1Â°C which will in turn change its resistance, by about 5%. This is referred to .as ..a self-heating error. It can be reduced by.,properly, .selecting the bridge voltage. ,. , The.: self-heat effect, was. restricted, to less than the 0<01Â°C allowable precision..,, Taking, a.reasonable .amount of error due to selfr-heating-to,.be.,50%,,oÂ£,. allowable precision or 0..005Â°G, the max--6 imum..power.â€žal.lowed. through, the. thermistor, is. 0.7 x 0.005 = 3.5 x 10 wat Taking an average value for the resistance of the thermistor to be 50,000fi, we get for the voltage across the thermistor a â€”6 3 value 3.5x10 x50xl0 or 0.42 volts. We can, therefore, use a maximum of 0.84 volts across the bridge. 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 GO Z LU LU LU cr cr LU (3 LL. LU LL- Q 1 4 HEAT TRANSFER RATE IN WATTS FIGURE 18. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEA'i iR^-S" RATE FOR 6.92 KV RMS BETWEEN THE CYLINDERS. FIGURE 19. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 7.91 KV RMS BETWEEN THE CYLINDERS. LU 1 2 3 4 5 6 HEAT TRANSFER RATE IN WATTS FIGURE 20. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 9.63 KV RMS BETWEEN THE CYLINDERS. 8 HEAT TRANSFER RATE IN WATTS FIGURE 21. DEPENDENCE OF TEMPERATURE DIFFERENCE ON HEAT TRANSFER RATE FOR 10.15 KV RMS BETWEEN THE CYLINDERS. LU O CO z: LLJ UJ LJJ cr cr LU o U_ LU U_ Q Q LU LU or 3 < cr UJ jfj 2 Lu z â€” â€” / â€” 10.80 KV â€” J. 1 I 1 1 1 1 1 2 3 4 5 6 HEAT TRANSFER RATE IN WATTS FIGURE 22. 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 far below the required precision. Error due to finite thickness., of .walls of the cylinders: The temperature of the inner cylinder was measured by the thermistors mounted on the inner wallâ€ž; The temperatures of the outer cylinder were those at the outer surface. For any true calculation of conductive heat transfer through the liquid, it is essential to correct for the finite wall thicknesses... The thicknesses of the walls were sufficiently small that it was estimated that the maximum error introduced is less than 0.5% and it was not considered necessary to make this correction. Errors due to .internal resistance of 1^, 1^ and V^: The sensitivity of the voltmeter. V^, was 20,0000/volt. The resistance of.the ammeters was 0.10 on 1A range compared to about'130.for the mid-section of the heating coil.. It was, therefore, found unnecessary to do any corrections for^the. internal resistance, of the meters. .3.5 Experimental Results. Experimental results, in the form.of.the temperature difference between.the cylinders .plotted against the heat .transfer rate in the central section of the annulus are shown in 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 linear, indicating & 9.63 KV o 10.15 KV Â© 1.3 _ Â© 10.80 KV 1.2 â€” 1.1 Â© 1.0 Â© Â© Â© I 1 i O & O O 1 1 1 1 500 1000 1500 2000 2500 3000 35.00 ELECTRICAL RAYLEIGH NUMBER FIGURE 23. NUSSELT NUMBER AS A FUNCTION OF ELECTRICAL RAYLEIGH NUMBER. 65 conductive heat transfer even for the maximum temperature differences attained. For potentials of 9.63, 10.15 and 10.80 kv rms, distinct breaks occur in the linear relationship indicating the onset of convective heat transfer. All the results are summarized in figure 23 as a plot of the Nusselt number v1(l1 - 12) ln(r2/ri) WU 4.1854 [2TTXL(T1 - T,,)] against the electrical Rayleigh number, R (equation 52, Chapter II). In the above equation x is the thermal conductivity of the fluid and L is the length of the measurement section between the points 8 and 10 (see figure 6). The values of viscosity and dielectric constant entering in the expression for R were taken at the mean of the temperatures of the cylinders. The thermal conductivity depends so slightly on the . temperature that its variation with temperature was ignored. Other constants used for the oil,are listed in Table 5. Figure 23 shows the experimentally determined critical electrical Rayleigh number for the onset of instability to be R = 2200 Â± 100. c 3.6 Accuracy The accuracy of the final results for Nu and R is estimated to be,Â±5% when all 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 for a range of voltages applied to the outer cylinder,, For lower voltages the plots are linear, indicating the heat transfer is by conduction alone. Above 7.91 kv rms, however, the plots show distinct breaks which occur at the onset of instability (Schmidt and Milverton, 1935). The electrical buoyancy force for small voltages was not sufficient to overcome thermal and viscous dissipation even for the maximum temperature differences created across the gap. However, if the temperature of the inner cylinder is raised further, we should expect appreciable convective heat transfer even for those voltages. The plot of the Nusselt number against the electrical Rayleigh number is shown in figure 23. In the conduction regime, the Nusselt number is found to have a slight downward trend from the expected value of unity. If there is a small temperature drop at the outer wall contact point during the initial run with no high voltage, this could lead to an underestimation of the temperature difference of figure 11. Consequently, the impressed temperature difference between the cylinders would be overestimated. A linear increase of this contact temperature difference with impressed temperature difference could result in a small decrease in the Nusselt number with increasing Rayleigh number. The value of the electrical Rayleigh number at which the onset of instability occurs is: R = 2200 Â± 100. c The theoretically determined value of R is 2.119.346 and compares 67 well with the experimental value. The disturbances which will be mani fested at marginal stability are characterized by the wave length A = ^ = ^=2.02d. k a c The value of a^ for this case is same as for the Rayleigh-Benard convection (ac=3.117). In the Rayleigh-Benard instability, the Nusselt number is a function of the Rayleigh number only. Here, however, the Nusselt number depends on the electrical Rayleigh number and the ratio of the inner cylinder radius to gap width. 4.2 Conclusions The present investigations have demonstrated the feasibility of constructing a laboratory model of thermal convection to simulate large scale geophysical flows that occur under a (nearly) central gravitational field. The marginal stability analysis of the flow was done by making several simplifying assumptions. The general validity of the principle of exchange of stabilities could not.be proved for this problem. However, the quality of the agreement of the experimental results with the numerically determined value (assuming the principle of exchange of stabilities) is a strong verification of the validity of the analysis. In constructing laboratory models of geophysical flows, we need to proceed to spherical geometry and rotation. We should also be interested in the actual form of the flow pattern as well as the values of various parameters at the onset of convection. All this will require the development of some new techniques, but in principle it now appears possible-68 REFERENCES Avsec, D. and Luntz^ M. 1937, Electricite Et Hydrodynamique - Quelques formes nouvelles des tourbillons electroconvections, C. R. Acad. Sci., Paris, 204, 757. Batchelor, G. K., 1954, Heat transfer by free convection across a closed cavity between vertical boundaries at different temperatures, Quart. Appl. 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Soc,, A, 176, 312-43. 70 Rayieigh, Lord, 1916, On convective currents in a horizontal layer of fluid when the higher temperature is on the underside, Phil. Mag., 6, 529-46. Rossby, C. G., 1941, The scientific basis of modern meteorology, Yearbook of Agriculture, U. S. Dept. of Agriculture, 599-655. Schmidt, R. J. and Milverton, S. W*, 1935, On the instability of a fluid when heated from below, Proc. Roy. Soc. (London), A, 152, 586-94. Silveston, P. L., 1958, Warmedurchgang in Waagerechten FlUssigk-eitsschichten, part 1, Forsch Ing. Wes., 24, 29-32 and 59-69. Smylie, D. E., 1966, Thermal convection in dielectric liquids and modelling in geophysical fluid-dynamics, Earth and Planetary Science Letters, 1, 339-40. Turnbull, R. Ji,1968, Electroconvective Instability with a stabilizing temperature gradient, Phys. Fluids, 11, 2588-96 and 2597-2603. Vening-Meinesz, F. .A., 1962, Thermal convection in the Earth's mantle, Continental Drift.edited by S. K. Runcorn, Academic Press. Vest, C. M. and Arpaci, V. S. , .1969, Stability of natural convection in a vertical slot,.J. Fluid Mech., 36, 1-16. 71 APPENDIX I TABLES OF THERMISTOR CALIBRATION 72 TABLE NO. 3 TABLE GF CAL IBRATICN POINTS FOR THERMISTORS i INNER CYLIND ER. ( 1 ) (2) ( 3) TEMPERATURE RESISTANCE RES ISTANCE RES I STANCE DEGREES C OH N S CHMS OHMS 15.037 79S12 7808a 72127 17.7 14 70364 68559 6351 7 20.450 61923 60155 55902 23. 166 54 6 71 52952 49352 25. 8<58 48310 46668 43625 28.646 42779 41194 38614 DIFFERENCE BETWEEN CALIBRATED RESISTANCE AND COMPUTED RESISTANCE. ( 1 ) ( 2 ) ( 3) TEMPERATURE RES IS T ANC E RES ISTANCE RES ISTANCE DEGREES C OHMS OHMS CHMS 15.C37 -0. 81 0.25 1.13 17.714 -G.88. -0.25 - 1.40 20.450 0.67 -0 .96 -0.86 23.166 6. 42 2.98 1 .39 25.898 -7.42 -2.51 0. 24 28.646 4 .10 0.74 -0.36 ( 1)THE RMIS TOR AT POINT 8 ON THE INNER CYLINDER. (2)THERMIST0* AT POINT 9 CN THE INNER CYLINDER. (3JTHERMISTCR AT POINT 1C ON THE INNER CYLINDER. 73 ABLE N TABLE POINT TEMPÂ£ DEGR 11 . 13 . 15. 17. 19 . ?1 . CF THE RATI EES 9 14 C 68 8 2 4 742 677 624 C L I 34ATI C.\ MV. ISTCWS. FCIMS FOR INLET AND CUT LEI ;KE C ( 1 ) ;< r S I S T i\C E CHrlS 9C481 a? 16 5 7 4 69 7 6 81 C4 6.? 08 2 56632 ( 2 i RES IS VANCE CH.MS 7 9 097 7 18 16 6 52 S3 59 505 54242 49 4 66 DIFFERENCE. BETWEEN CALIBRATED RESISTANCE AND COMPUTED RESISTANCE. TEMPERATURE DEGREES C 11.9 14 1 3 .868 15.024 17.742 19 .677 21 .624 ( 1 ) RESISTANCE OHMS 3.4 4 -6 , -0, 7. -4, 0, 6.1 44 Cc 19 8,3 (2) RES ISTANCE OHMS 3. 69 -5.25 2.0 6 2.70 -0 .46 -0 .53 (1) T HE RMIS TOR AT PCINT 11 AT THE INLET POINT TC THE THERMAL FLASK. (2) THERKISTGK AT FC I NT 12 AT THE OUTLET POINT TO THE THERMAL F LASK. \ 74 APPENDIX II TABLES OF HEAT TRANSFER MEASUREMENTS 75 TABLE NO. "6 VOLTAGE BETWEEN THE C YLINDER S = 0.0 KV (1) (2) {J ) (4) (5) (6) (7) (8) 16. 34 1 6.35 16. 3 5 16 .61 16.82 0.0 0.0 0.0 17.58 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. 83 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 2 4 . 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 27. 83 16 .64 16.80 0.787 0.201 7.97 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 II IN AMPERES. (7) . CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VOLTS. 76 TABLE NG. 7 VOLTAGE BETWEEN THE CYLINDERS3 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 25 .5 1 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 29.33 29.33 29.34 16.78 (1) TEMPERATURE OF INNER CYLINDER AT POINT 10. (2) TEMPERATURE CF INNER CYLINCER AT POINT 9. (3) TEMPERATURE OF INNER CYLINDER AT PCINT 8. (4) TEMPERATURE OF CIRCULATING OIL AT INLET POINT 11. v{ 5 ) TEMPERATURE OF CIRCULATING OIL AT OUTLET POINT 12. (6) CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VOLTS. 77 TABLE NO. B VOLTAGE BETWEEN THE CYL INDERS= 6 .ODKtf (i) ( 2) (3) (4) (5) (6) (7) (8 ) 19.49 19.50 19 .50 16.86 17.08 0.437 0. 139 4. 03 20. 4 4 20.45 20.45 16. 80 17.02 3.496 0.156 4.61 21 .36 21.37 21.37 16. 87 17. 15 .0.552 0. 174 5.13 22. 35 22.35 22 .34 16.8 7 17.16 0.602 0. 189 5.58 23. 26 23.26 2 3.25 16. 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 25. 54 25.55 25. 53 16. 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) TEMPERATURE OF INNER CYLINDER AT POINT 10. (2) TEMPERATURE CF INNER CYLINDER AT POINT 9. (3) TEMPERATURE CF INNER CYLINDER AT PCINT 8. (4) TEMPERATURE CF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE OF CIRCULATING OIL AT OUTLET POINT 12. (6) CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) VCLTAGE VI IN VOLTS. 78 TABLE NO. 9 VOLTAGE BETWEEN THE CYLINDERS= 6.92KV '(I). (2) (3) (4) (5) lb) ' il) (8) 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 21. 28 21. 28 21. 26 16. 75 17.04 0.547 0.1 73 5.08 22.25 22 .25 22 .24 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 .38 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) TEMPERATURE GF INNER CYLINDER AT POINT 10. (2) TEMPERATURE CF INNER CYLINDER AT PCINT 9. (3) TEMPERATURE OF INNER CYLINDER AT PCINT 0. (4) TEMPERATURE CF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE CF CIRCULATING OIL AT OUTLET POINT 12. (6) CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VOLTS. 79 TABLE NO. 10 VOLTAGE BETWEEN THE CYLINDERS= 7.91KV (I) ( 2 ) (3 ) (4) (5 ) (6) (7) (8) 19. C4 1 9. 04 19.04 16.66 16.92 0.402 0.128 3.72 19.75 19.75 19.75 16. 67 16. 89 0.452 C. 139 4.23 20.62 2 0.62 20.61 16.70 16.86 0.509 0.155 4.78 21. 6T 21. 68 21. 67 16. 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 16. 74 16. 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 16. 64 16. 82 0.895 0. 291 8.20 (1) TEMPERATURE CF INNER CYLINCER AT PC I NT 10. (2) TEMPERATURE OF INNER CYLINDER AT POINT 9. (3) TEMPERATURE OF INNER CYLINDER AT POINT 8. (4) TEMPERATURE CF CIRCULATING OIL AT INLET POIMT 11. (5) TEMPERATURE OF CIRCULATING OIL AT CUTLET POINT 12. (6) CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VCLTS. 80 TABLE NO. 11 VOLTAGE BETWEEN THE CYLINDERS* 9.63KV (1 ) (Z ) (3 ) (4) (5) (6) (7) (3) 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) TEMPERATURE OF INNER CYLINCER AT POINT 10. (2) TEMPERATURE OF INNER CYLINCER AT PCINT 9. (3) TEMPERATURE OF INNER CYLINDER AT POINT 8. (4) TEMPERATURE OF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE CF CIRCULATING OIL AT OUTLET POINT 12. (6) CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) 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 0. 103 3.2 0 18.9 2 13.92 10 .9 2 16 .69 16.82 0.391 0. 119 3.65 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 0. 851 C.281 7.75 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) ' TEMPERATURE CF INNER CYLINCER AT PC I NT 10. (2) TEMPERATURE OF INNER CYLINDER AT POINT 9. (3) TEMPERATURE OF INNER CYLINDER AT POINT 8. (4) TEMPERATURE CF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE CF CIRCULATING 01L AT CUTLET POINT 12. (6) CURREMT II IN AMPERES. (7) CURRENT 12 IN AMPERES. (8) VOLTAGE VI IN VOLTS. 82 TABLE NO. 13 VOLTAGE BETWEEN THE CYLINDERS =, 10. 80KV (1 ) (2 ) (3 ) (4) (5) (6) ( 7) (8) 17.S4 17.95 17. 95 16.66 16.76 0.310 0.095 2.88 18. Al 18.A2 18.42 16.61 16.77 0.354 0.108 3.30 19. C9 "19.10 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 16. 57 16. 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 26. 10 26. 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 16. 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) TEMPERATURE OF INNER CYLINDER AT POINT 10. (2) TEMPERATURE CF INNER CYLINCER AT POINT 9. (3) TEMPERATURE CF INNER CYLINDER AT PCINT 8. (4) TEMPERATURE OF CIRCULATING OIL AT INLET POINT 11. (5) TEMPERATURE CF CIRCUL AT ING OIL AT OUTLET POINT 12. (6) . CURRENT II IN AMPERES. (7) CURRENT 12 IN AMPERES. ( (8) VCLTAGE VI IN VOLTS. 83 APPENDIX III TABLES OE CORRECTED DATA 84 TABLE NO 14 VOLTAGE BETWEEN THE CYLINDERS= 0.0 KV (1) (2) (3) (4) (5) 16.35 16. 3 6 C. 0 0.0 . 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) MEAN TEMPERATURE DF INNER CYLINDER. (2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CORRECTED 12. (5) CORRECTED VI. 85 TABLE NO 15 VOLTAGE BETWEEN THE CYL IN DEK S= 4.06 KV ( 1) 12) (3) (4) (5) 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) MEAN TEMPERATURE OF INNER CYLINDER. (2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CCRRECT ED 12. (5) CORRECTED VI. 86 TABLE NG 16 VOLTAGE BE T-WEEN THE CYLINOERS= 6.00 KV (1) {2 ) (3 ) (4) (5) 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) NEAN TEMPERATURE OF INNER CYLINDER. <2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CORRECT ED 12. (5) CORRECTED VI. 87 TAnLE KO 17 VOLTAGE BETWEEN THE CYLINDE RS = 6. 92 KV ( 1 ) (2 ) (3 ) (4) (5) 19.05 17.48 C.4068 0. 1280 3.780 19.67 17.74 0.4-506 0. 1430 4. 183 2 0.44 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 19. 08 G.6539 0. 2069 6.097 24.21 19 .36 0.6994 0. 2246 6. 461 25.33 19.81 0.7463 0 . 2316 6.852 26.91 20.4G C. 8142 0. 2613 7. 544 28 .38 21 .07 0.8684 0. 2753 8. 100 29.44 21.49 0.9128 0. 2 913 3.500 Q) MEAN TEMPERATURE OP INNER CYLINDER. (2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CORRECTED 12. (5 ) CORRECTED VI . 88 T A PL E NO 18 VOLTAGE BETWEEN THE CYLINDER$= 7.91 KV (1) (2) (3) (4) { 5) 19. 04 1 7. 43 C.4049 0.1300 3.747 19.7 5 17.69 C.4546 0. 1410 4. 247 20.62 10 .03 0.5112 0.1568 4. 788 21.67 18.43 C. 5613 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.76 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) MEAN T.EMPERATURE OF INNER CYLINOER. (2) MEAN TEMPERATURE OF OUTER CYLINCER. (3) CORRECTED II. (4) CORRECTED 12. <5) CCRRECTED VI. 89 TACLE Nil 19 VOLTAGE BETWEEN THE CYLINDERS= 9.63 KV (1) (2) (3) (4 ) (5) 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) MEAN TEMPERATURE OF INNER CYLINOER. (2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CORRECTED 12. (5) CORRECTED VI. 90 TABLE NO 2 0 VOLTAGE BETWEEN THE CYLINDERS= 10.15 KV (1) ; (2) (3) (4) 15) 17.75 16.93 0.2906 C.Q8C8 2.682 18. 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. 5093 C.1628 4.790 21 .20 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 19. 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 .67 0.9477 0.2943 8.850 11) MEAN TEMPERATURE OF INNER CYLINDER. (2) MEAN TEMPERATURE OF OUTER CYLINDER. (3) CORRECTED II. (4) CORRECTED 12. (5) CORRECTED VI. 91 1ABLF: NO 21 VOLTAGE BETWEEN THE CYLINDERS= 10.80 KV ( 1) (2 ) (3 ) (4) ( 5 ) 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) MEAN TEMPERATURE OF INNER CYLINDER. (2) F E AN TEMPERATURE OF OUTER \CYLINDER. 13) CORRECTED II. . (4) CORRECTED 12. (5) CORRECTED VI.
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A laboratory study of thermal convection under a central force field Chandra, Bhuvanesh 1971
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Title | A laboratory study of thermal convection under a central force field |
Creator |
Chandra, Bhuvanesh |
Publisher | University of British Columbia |
Date | 1971 |
Date Issued | 2011-03-14 |
Description | This thesis presents the results of a theoretical and experimental study of thermal convection under the influence of a central force field. Flows in the atmosphere and in the core of the earth are thought to occur under a near balance between Coriolis and buoyancy forces. Thus, a desirable model of these flows would include spherical symmetry in the force field and rotation. The present study, in which convection under a central force field in cylindrical geometry has been achieved, is the first step towards such a model. The system consists of a cool outer cylinder and a hot inner cylinder with a dielectric liquid (silicone oil) filling the annulus between them. The common axis of the cylinders is vertical. The inner cylinder is grounded and the outer cylinder is kept at a high alternating (60 Hz) potential. This intense alternating electric field provides the radial buoyancy force which results in convective heat transfer at a certain critical temperature gradient. The fluid in the system is found to behave like a layer of fluid in a gravitational field, heated from below. Below a certain critical value of a dimensionless number (equivalent to the Rayleigh number with the electrical force substituted for gravity) there is no convective heat transfer. Above the critical value, flow sets in with the convective heat transfer proportional to the modified Rayleigh number. Marginal stability analysis gives a critical electrical Rayleigh number in agreement with the experimentally determined value. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-03-14 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0052446 |
Degree |
Doctor of Philosophy - PhD |
Program |
Geophysics |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/32441 |
Aggregated Source Repository | DSpace |
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