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MIT OpenCourseWare http://ocw.mit.edu Continuum Electromechanics For any use or distribution of this textbook, please cite as follows: Melcher, James R. Continuum Electromechanics. Cambridge, MA: MIT Press, 1981. Copyright Massachusetts Institute of Technology. ISBN: 9780262131650. Also available online from MIT OpenCourseWare at http://ocw.mit.edu (accessed MM DD, YYYY) under Creative Commons license Attribution-NonCommercial-Share Alike. For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.10Electromechanics with Thermaland Molecular DiffusionQ-i-:I:10.1 IntroductionThe general three-way coupling between electromagnetic, mechanical and thermal or molecular sub-systems might be pictured as in Fig. 10.1.1. Thermal interactions are the subject of the first halfof this chapter while the second is concerned with the molecular subsystem.Diffusion dynamics is familiar from the mag-netic diffusion of Chap. 6 and the viscous diffusionof Chap. 7. For both thermal and neutral moleculardiffusion processes, Sec. 10.2 builds on this back-ground by identifying the characteristic times,lengths and dimensionless numbers with analogousparameters from these previous dynamical studies.Much of the sinusoidal steady-state and transientdynamics, boundary layer models and transfer rela-tions are equally applicable here.Electrical heating and the need for conduc-tion and transport of that heat is often crucialin engineering problems. Section 10.3 is there-f- ore d4 evoLte d4 to t U al one-way acoup l n 1*- n .oe Uihheatgenerated electrically in a volume is removed bythermal diffusion, (a) in Fig. 10.1.1. The three- Fig. 10.1.1. Three-way coupling.way coupling illustrated in Sec. 10.4 involves anelectrical conductivity that is a function of temperature, (b) in Fig. 10.1.1, an electric force createdby the resulting property inhomogeneity, (f), and a convection that contributes to the heat transfer,(d).The rotor model introduced in Sec. 10.5 should incite an awareness of analogies with dynamicalphenomena encountered in Chaps. 5 and 6 on circulating fluids, but it should not be forgotten that thediffusion phenomena discussed in many of these sections also occur in solids. The magnetic-field-stabilized Bsnard type of instability discussed in Sec. 10.6 is an example of a continuum phenomenathat might be modeled by the rotor. This study gives an opportunity to illustrate how the Rayleigh-Taylor types of instability from Chap. 8 are modified if property gradients have their origins inthermal or molecular diffusion.Because the effect of molecular diffusion of neutral species is similar to that of thermal con-vection, the sections on molecular diffusion are confined to the diffusion of charged species. Dif-fusional charging of small macroscopic particles subjected to unipolar ions is the subject of Sec. 10.7.Section 10.8 is aimed at picturing the standoff between diffusion and migration that makes a doublelayer possible. Based on this simple model, shear-flow electromechanics are modeled in Sec. 10.9 andused to introduce electro-osmosis and streaming potential as electrokinetic phenomena. Another electro-kinetic phenomenon, electrophoresis of particles, is taken up in Sec. 10.10. Sections 10.11 and 10.12introduce electrocapillary phenomena, where the double-layer surface force density from Sec. 3.11 comesinto play. Sections 10.7 and 10.8 involve links (a) and (b) in Fig. 10.1.1, while Secs. 10.9, 10.10and 10.12 involve all links. The sections on molecular diffusion suggest the scale and nature of elec-tromechanical processes found in electrochemical, biological and physiological systems.10.2 Laws, Relations and Parameters of Convective DiffusionThermal Diffusion: The most common thermal conduction constitutive relation between heat flux andtemperature is Laplace's law:= -k VT (1)where kT is the coefficient of thermal conductivity. Not only in a perfect gas, but also for manypurposes in a liquid, the internal energy is usefully taken as proportional to the temperature. Thus,the energy equation, E'q. 7.23.4, becomesBT + 2 d (a- + v.VT ~= T+ ý • d d; 'f +T -pV~v (2)PCvwhere the thermal diffusivity is defined as KT kT/pc,. From left to right, terms in this expressionrepresent the thermal capacity, convection and conduction. The last term is due to electrical andviscous dissipation and power entering the thermal system because of dilatations. Although cv and kTare in general functions of temperature, thermally induced variations of other parameters are usuallymore important and so cvand kT have been taken as constant in writing Eq. 2.10.1Secs. 10.1 & 10.2Table 10.2.1. Thermal diffusion parameters for representative materials.Temp. Mass Specific Thermal Thermal PrandtlMaterial (OC) density heat conductivity diffusivity numberP (kg/m3) (J/kgoC) kT (watts/moK) KT (m2/s) PT PTLiquid cWater 10 1.000x103 4.19x103 0.58 1.38x107 9.5" 30 0.996x103 4.12x103 0.61 1.46x107 5.570 0.978x103 3.96x103 0.66 1.61x10-7 2.6" 100 0.958x103 3.82x103 0.67 1.66x10- 7 1.87 Glycerine 10-70 1.26x103 2.5x103 0.28 0.89x10-1.3x10Carbon tetra- 3 3 0.832xi0- 7 7.315 1.59x10 0.83x10 0.11 0.832x10 7.3chloride3 3 4.2x10-6 2.7x10- 2hercury 20 13.6x100.14x108.0 CErelow-117 50 8.8x103 0.15x103 16.5 1.25x10- 5 ,-5xlO- 3Gases cvAir 20 1.20 0.72x103 2.54x10- 2 2.1x10- 5 0.72100 0.95 0.72x103 3.17x10- 2 3.3x10-5 0.70Solids CPAluminum 25 2.7x10 0.90x103 240 9.4x1 -Copper 25 8.9x103 0.38x103 400 llx10 7Vitreous quartz 50 2.2x103 0.77x103 1.6 9.4x1 -71oWith electrical and viscous heating given, and work done by dilatations negligible (as isusually the case in liquids), Eq. 2 becomes a convective diffusion equation analogous to magneticdiffusion equations in Chap. 6 and viscous diffusion equations in Secs. 7.18-7.20. Instead of themagnetic or viscous diffusion times, the thermal diffusion timeTT = a 2/Kcharacterizes transients having A as a typical length. For processes determined by convection, it isthe ratio of this thermal diffusion time to the transport time, k/u, that is relevant. With u atypical fluid velocity, this dimensionless number is defined as the thermal Peclet number,RT= u/KTThe response to sinusoidal steady-state thermal excitations with angular frequency w is likely to havea spatial scale that is much shorter than other lengths of interest, in which case the thermal diffusionskin depth_2K;6, =1wis the length over which the thermal inertia


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