Materials Engineering (MED 123)

Transcription

1 Materials Engineering (MED 123) Igor Medved Department of Materials Engineering and Chemistry, Czech Technical University, Prague MAEN 123 p. 1

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3 In general, heat is transferred by one or more of the following mechanisms: conduction convection radiation These are known from everyday experience When we warm our homes in the winter months, heat is transferred to the inside walls by convection, through the walls by conduction, and away from the outside walls to the surroundings by convection and possibly radiation When cooked food is taken out of the oven, heat is transferred to the surroundings initially by radiation and convection from the outside surface, and the cooling of the outer part so affected causes conduction of heat from the center to the outside with continued removal from there to the surroundings In the following we will be concerned only with heat conduction A couple of books to consult: J. C. Jones, Principles of Thermal Sciences and their Application to Engineering. Whittles Publishing, 2. M. N. Ozisik, Heat Conduction. John Wiley & Sons, MAEN 123 p. 3

4 In gases heat is transferred via collisions between molecules of the gas A molecule of the gas at the temperature T undergoes translational motion with an average kinetic energy given as ε = 3 2 k BT, wherek B = JK 1 is the Boltzmann constant AtT = 3 K the energy isε = J Since the kinetic energy is equal to 1 2 mv2, the energyεcorresponds to the molecular speed 2ε v m ms 1 (the mass of an oxygen moleculem 1 26 kg) In liquids the translation of molecular groups and the accompanying thermal energy transfer are the origin of heat conduction contrary to gases, a liquid molecule is never in isolation from others MAEN 123 p. 4

5 There are two basic mechanisms of heat conduction in : by lattice vibrations and by means of the thermal energy of any delocalized electrons present The presence of delocalized electrons in metals and their absence from non-metals (such as ceramics or wood) is the basic reason why metals are good heat conductors, while non-metals are thermal insulators The electrons are treated as a gas of molecules (an electron gas ) each having a thermal energy A metal is viewed as a vibrating lattice of positive ions in a sea of electrons An electron loses its thermal energy by colliding not with another electron but with one of the positive ions constituting the lattice Why is it that heat conduction is so much faster in metals than in gases? Since the average speed is very much greater in an electron gas than in an actual gas At the same thermal energyεthe average speed isv ε/m, so that v electron gas moxygen v oxygen gas m electron Thus, the average speed in an electron gas is about2 times higher than in an oxygen gas MAEN 123 p.

6 states that the heat fluxq (=the amount of heat transferred through unit area per unit time) is proportional to the temperature gradient, the constant of proportionality being the thermal conductivity λ (also denoted as k) So, whenever there is a temperature difference T between two places, heat flows from the hotter to the colder place The higher T, the larger the heat fluxq The larger the difference x between the places, the smaller the heat fluxq MAEN 123 p. 6

7 In one dimension the mathematical formulation of the law is q = λ T x or, more precisely, q = λ dt dx (I.1) The negative sign arises from the fact that the heat flows in the direction opposite to the increase int q has the unitjm 2 s 1 = Wm 2 ;λhas the unitwm 1 K 1 In three dimensions the mathematical formulation of the law is q = λ T (I.2) Heat flux is a vector (it has a magnitude and a direction) T = gradt is the gradient of temperature; it is a vector pointing in the direction of the greatest rate of temperature increase In the rectangular coordinatesx,y,z it is given as T = x + y + z MAEN 123 p. 7

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10 The thermal conductivity is an important thermal property of materials, it characterizes the amount of heat transfer rate per unit area and per unit temperature gradient Any substance a solid, liquid, or gas can be assigned a numerical value of thermal conductivity; however, the value changes with temperature and pressure Of course, in heat transfer within (= gases and liquids) the dominant mode of heat transfer might be convection It is quite acceptable to extend the concept of thermal conductivity to powdered, fibrous, or shredded media, such as a bed of sawdust or a bale of wool Each of these is actually two-component, comprising the particles and the air voids (pores), but a single value of the thermal conductivity can be taken to apply For a specified temperature difference T and thickness x, the rates of heat transfer (= heat flow) will differ according to λ of the conducting material Let a material of thickness x = 2 cm have one side kept at 7 C and the other side at 2 C If the material is copper (Cu), thenλ = 386Wm 1 K 1 and the heat flow is q = = 87Wm 2 For stainless steel (λ = 1Wm 1 K 1 ) we getq = 34Wm 2 For cardboard (λ =.7Wm 1 K 1 ) we get onlyq = 16Wm 2 MAEN 123 p. 1

11 Typical range ofλof various materials + the effect of the temperature onλ MAEN 123 p. 11

12 Expresses the conservation (balance) of energy in a given material Consider a volume within the material, then Time change in internal energy (energy stored) in If is very small, then = Heat flow into through its surface Rate of energy + generation in Time change in internal energy (energy stored) in = U t = m c p T t = ρ c p T t Heat flow into through its surface = q na MAEN 123 p. 12

13 Energy generation may occur due to processes that take place in the material (such as phase changes or chemical, electrical, nuclear, and other sources) Rate of energy generation = amount of heat produced by such processes per unit volume and unit time = denoted as, say,g(x,t) that may depend on position and/or time (unitsjs 1 m 3 = Wm 3 ) Rate of energy generation in a small volume is equal tog MAEN 123 p. 13

14 If is not necessarily small, then Time change in internal energy (energy stored) in Heat flow into = ρ c p T t through its surface A = q na q nda Rate of energy generation in = g gd ρc p t d The energy balance then gives ρc p t d = A q nda+ gd To proceed further, we need to convert the surface integral (the second term) into a volume integral This is possible thanks to J. C. F. Gauss (German mathematician, ): A q nda = ( q)d, q = div q is the divergence of the heat flux = the heat flow from an infinitely small volume around a given point In the rectangular coordinatesx,y,z it is given as q = q x x + q y y + q z z if the vector q = (q x,q y,q z ) MAEN 123 p. 14

15 The energy balance now reads ρc p t d = ( q)d + gd Further manipulations Using q = λ T, we get ρc p t d = λ or ( T)d + {ρc p t λ( T) g } d = The latter equality must be true for an arbitrary volume This is possible only if the expression in{...} is equal to Finally, the heat equation reads gd ρc p t = λ T g T = T is called the Laplacian oft In the rectangular coordinatesx,y,z one has T = 2 T x T y T z 2 MAEN 123 p. 1

16 If no generation occurs (g = ), then the heat equation simplifies to ρc p t = λ T i.e. t = a T a = λ ρcp = thermal diffusivity (unitm2 s 1 ); sometimes denoted asα Important thermal property of materials (depends on the temperature) It is associated with the speed of propagation of heat into the body during temperature changes (the higher a, the faster the propagation) For example, consider a rod at room temperature and cool one of its ends to, say,k. How fast will the temperature of rod drop to its half at, say,3 cm from the cooled end? MAEN 123 p. 16

17 The heat equation cannot be solved without knowing how the material is heated or cooled Usually we knowt within the body (at all positionsx,y,z) at some instantt (mostly the beginning of the measurement) = initial condition: T(x,y,z,t ) = a known function ofx,y,z We must also know what happens witht on the surface (boundary) of the material = boundary Temperature on the boundary is prescribed at all times: T(x,y,z,t) = a known function for any position on the boundary (for example,t is constant on the boundary = the surface of the material is heated/cooled at a prescribed temperature) Heat flow is prescribed on the boundary at all times: n = a known function for any position on the boundary, where the derivative is along the outward normal to the surface (for example, there is no heat flow into the body, i.e., n = ) Other forms of boundary may apply in a given situation MAEN 123 p. 17