Thermal Properties of materials

Transcription

1 Thermal Properties of materials Introduction By thermal property is meant the response of a material to the application of heat. As a solid absorbs energy in the form of heat its temperature rises and its dimensions increase. The energy may be transported to cooler regions of the specimen if temperature gradients exist, and ultimately, the specimen may melt. Heat capacity, thermal expansion, and thermal conductivity are properties that are often critical in the practical utilization of solids. Heat capacity Heat capacity in metals (Vibrational heat capacity/specific heat) A solid material, when heated, experiences an increase in temperature signifying that some energy has been absorbed. Heat capacity is a property that is indicative of a material s ability to absorb heat from the external surroundings; it represents the amount of energy required to produce a unit temperature rise. In mathematical terms, the heat capacity C is expressed as follows: C = dq/ dt Where dq is the energy required to produce a dt temperature change. Heat capacity is specified per mole of material (e.g., J/mol-K, or cal/mol-k). Vibrational Heat Capacity: In most solids the principal mode of thermal energy assimilation is by the increase in vibrational energy of the atoms. Again, atoms in solid materials are constantly vibrating at very high frequencies and with relatively small amplitudes. Rather than being independent of one another, the vibrations of adjacent atoms are coupled by virtue of the atomic bonding. These vibrations are coordinated in such a way that travelling lattice waves are produced, a phenomenon represented in below Fig. These may be thought of as elastic waves or simply sound waves, 1

2 having short wavelengths and very high frequencies, which propagate through the crystal at the velocity of sound. The vibrational thermal energy for a material consists of a series of these elastic waves, which have a range of distributions and frequencies. Only certain energy values are allowed (the energy is said to be quantized), and a single quantum of vibrational energy is called a phonon. (A phonon is analogous to the quantum of electromagnetic radiation, the photon.) On occasion, the vibrational waves themselves are termed phonons. The thermal scattering of free electrons during electronic conduction is by these vibrational waves, and these elastic waves also participate in the transport of energy during thermal conduction. Fig 4: Schematic representation of the generation of lattice waves in a crystal by means of atomic vibrations Heat capacity of solids: Classical theory, Einstein and Debye theories of Specific heat. (Board Teaching & PPT) The heat capacity per unit mass and has various units (J/kg-K, cal/g-k) There are really two ways in which this property may be measured, according to the environmental conditions accompanying the transfer of heat. One is the heat capacity while maintaining the specimen volume constant C v, the other is for constant external pressure, which is denoted by C p. The magnitude of C p is almost always greater than C v, however, this difference is very slight for most solid materials at room temperature and below. Temperature Dependence of the Heat Capacity 2

3 The variation with temperature of the vibrational contribution to the heat capacity at constant volume for many relatively simple crystalline solids is shown in Fig 5 is zero at 0 K, but it rises rapidly with temperature; this corresponds to an increased ability of the lattice waves to enhance their average energy with ascending temperature. At low temperatures the relationship between and the absolute temperature T is where A is a temperature-independent constant. Above what is called the Debye temperature levels off and becomes essentially independent of temperature at a value of approximately 3R, R being the gas constant. Thus even though the total energy of the material is increasing with temperature, the quantity of energy required to produce a one-degree temperature change is constant. The value of θ D is below room temperature for many solid materials, and 25 J/mol-K is a reasonable room-temperature approximation for C v. Fig 5: The temperature dependence of the heat capacity at constant volume;is the Debye temperature. Applications of specific heat capacity: Before going to know the applications, first of all we should able to know the characteristics of an object with low and high specific heat. Characteristics of an object with low specific heat: - Fast heated up: have a faster temperature increase - Fast cooled down: have a faster temperature decrease - Sensitive to temperature changes 3

4 - Ex: Aluminium, copper etc Applications: 1) Substances having a low specific heat capacity, are very useful as material in cooking instruments such as frying pans, pots, kettles and so on, because, they can be quickly heated up even when small amount of heat is supplied. 2) Sensitive thermometers also must be made from materials with low specific heat capacity so that it can detect and show a change of temperature rapidly and accurately. Characteristics of an object with high specific heat: - Heats up and cools down at a slower rate - Requires more heat to rise its temperature by a specific amount - Can absorb a great amount of heat - Ex: plastic, water, concrete etc Applications: 1. Substances that have a high specific heat capacity is suitable as a material for constructing kettle handlers, insulators and oven covers, because, a high amount of heat will cause only a small change in temperature aka the material won't get hot too fast! 2. Heat storage instruments are very useful and they are usually made of substances with a high specific heat capacity. 3. Water as a cooling agent acts excellent as a cooling agent in engines. Water is also used in houses in cold climate countries because as it is heated up (boiled) it tends to retain heat and warm the house due to its high specific heat capacity. Thermal Expansion and Thermal Conductivity in Metals, Ceramics and Polymers 4

5 Thermal expansion in metals, ceramics and polymers. Most solid materials expand upon heating and contract when cooled. The change in length with temperature for a solid material may be expressed as follows: l f l o l o =α i (T f T o ) Where l o and l f represent, respectively, initial and final lengths with the temperature change from T 0 to T f. The parameter α i is called the linear coefficient of thermal expansion; it is a material property that is indicative of the extent to which a material expands upon heating. Heating or cooling affects all the dimensions of a body, with a resultant change in volume. Volume changes with temperature may be computed as ΔV/V o = α v ΔT Where ΔV and V o are the volume change and the original volume, respectively, and α v symbolizes the volume coefficient of thermal expansion. In many materials, the value of is anisotropic; that is, it depends on the crystallographic direction along which it is measured. For materials in which the thermal expansion is isotropic α v = 3 α i. Fig 6: Plot of potential energy versus interatomic distance From an atomic perspective, thermal expansion is reflected by an increase in the average distance between the atoms. This phenomenon can best be understood by consultation of the potential energy-versus-inter atomic spacing curve for a solid material introduced in above figure. The curve is in the form of a potential energy trough, and the equilibrium interatomic 5

6 spacing at 0 K, corresponds to the trough minimum. Heating to successively higher temperatures (T 1, T 2 and T 3 etc) raises the vibrational energy from E 1 to E 2 to E 3 and so on. The average vibrational amplitude of an atom corresponds to the trough width at each temperature, and the average interatomic distance is represented by the mean position, which increases with temperature from r 0 to r 1 to r 2 and so on. Thermal expansion is really due to the asymmetric curvature of this potential energy trough, rather than the increased atomic vibrational amplitudes with rising temperature. If the potential energy curve were symmetric (Figure), there would be no net change in interatomic separation and, consequently, no thermal expansion. For each class of materials (metals, ceramics, and polymers), the greater the atomic bonding energy, the deeper and more narrow this potential energy trough. As a result, the increase in interatomic separation with a given rise in temperature will be lower, yielding a smaller value of the linear coefficients of thermal expansion for several materials. With regard to temperature dependence, the magnitude of the coefficient of expansion increases with rising temperature Thermal expansion in metals, ceramics and polymers: a. Metals The linear coefficients of thermal expansion for some of the common metals range between about 5 x 10-6 and 25 x 10-6 ( C) -1. For some applications, a high degree of dimensional stability with temperature fluctuations is essential. This has resulted in the development of a family of iron-nickel and iron-nickel-cobalt alloys that have α l values on the order of 1 x 10-6 ( C) -1. One such alloy, tradename of Kovar has been designed to have expansion characteristics close to those of borosilicate (or Pyrex) glass; when joined to Pyrex and subjected to temperature variations, thermal stresses and possible fracture at the junction are avoided. Kovar and two other low-expansion alloys (Invar and Super-Invar) that have very small α l values. b. Ceramics Relatively strong inter-atomic bonding forces are found in many ceramic materials as reflected in comparatively low coefficients of thermal expansion; values typically range between about 0.5 x 10-6 and 15 x 10-6 ( C) -1. For non-crystalline ceramics and also those having cubic crystal structures, α l is isotropic. Otherwise, it is anisotropic; and, in fact, some ceramic materials, upon 6

7 heating, contract in some crystallographic directions while expanding in others. For inorganic glasses, the coefficient of expansion is dependent on composition. Fused silica (high-purity SiO 2 glass) has a small expansion coefficient, 0.4 x 10-6 ( C) -1. This is explained by a low atomic packing density such that inter-atomic expansion produces relatively small macroscopic dimensional changes. Ceramic materials that are to be subjected to temperature changes must have coefficients of thermal expansion that are relatively low, and in addition, isotropic. Otherwise, these brittle materials may experience fracture as a consequence of non-uniform dimensional changes in what is termed thermal shock. c. Polymers Some polymeric materials experience very large thermal expansions upon heating as indicated by coefficients that range from approximately 50 x 10-6 to 400 x 10-6 ( C) -1. The highest α l values are found in linear and branched polymers because the secondary intermolecular bonds are weak, and there is a minimum of cross linking. With increased cross linking, the magnitude of the expansion coefficient diminishes; the lowest coefficients are found in the thermosetting network polymers such as phenol-formaldehyde, in which the bonding is almost entirely covalent. Applications of thermal expansion: 1. Fitting of parts The major application of this physics phenomenon is fit parts over one another. Let us understand it with a simple example a bushing can be fitted over a shaft by making its inner diameter slightly smaller than shaft s diameter.after then it is heated until it fits over the shaft and on cooling it makes a tight fit. The wheels of rolling-stock, specially driving wheels of locomotives, are used to be fitted with steel tyres and make this fit a tight one, tyre is made slightly smaller in diameter than the original diameter of the wheel.and before being fitted the tyre is heated uniformly by gas burners.this results in expansion that enables the tyre to be slipped easily over the wheel.after sometime when it get cooled, it makes a tight fit on wheels.so for tight-fitting of steel over the wheels, concepts pf thermal expansion is very useful. 7

8 2. Riveting This concept is also beneficial in case of riveting together the steel plates and girders used in shipbuilding and other construction work. The rivets are first heated as heating softens them and they can be are easily burred into a head by pneumatic hammers.after that when contraction occurs, plates get pulled together tightly. 3. Watches and Thermostats Due to ability of invar (An alloy of steel and nickel) to show exceptionally small expansion even when heated to high temperature it has many practical applications.for example, Invar is used in used in watches and thermostats. 4. Jar lids This is one of the most common example of thermal expansion.we often use to keep the metal lid of glass container. The reason behind it is that the high-temperature water causes the expansion of metal lid whereas Glass being a bad conductor of heated and also having a low coefficient of expansion remains unaffected. 5. Hot Air Balloons Other example of this phenomenon is Hot-air balloons. It is practical use of the practical use of the thermal expansion difference between a gas and a solid. In this case the hot air inside the balloon bag expands more quickly than the outside container therefore it stretches the bag so that it expands and displaces the colder (heavier) air outside the bag. This difference between lower densities of air inside the bags compare to the lower density of air outside the bag is the main reason for hot air balloon to rise. Similarly cooling the air inside the bag makes the balloon to come down. 6. Thermometers This device is another application of thermal expansion. Most of the thermometers contain a liquid (usually alcohol or mercury).this liquid is constrained to flow in one direction only (along the tube) due to changes in volume that are caused because of temperature change. 7. Other interesting and practical examples of thermal expansion 8

9 The material developed for filling teeth has the same expansion as the natural enamel of the tooth and the steel which is used to reinforce concrete has the same expansion as that of concrete. Process of thermal conduction in solids Thermal Conductivity Thermal conduction is the phenomenon by which heat is transported from high to low temperature regions of a substance. The property that characterizes the ability of a material to transfer heat is the thermal conductivity. It is best defined in terms of the expression. Q= K dt dx where q denotes the heat flux, or heat flow, per unit time per unit area (area being taken as that perpendicular to the flow direction), k is the thermal conductivity, and is the dt/dx temperature gradient through the conducting medium. Mechanisms of Heat Conduction: Heat is transported in solid materials by both lattice vibration waves (phonons) and free electrons. A thermal conductivity is associated with each of these mechanisms, and the total conductivity is the sum of the two contributions, k=k l +k e Where k l and k e represent the lattice vibration and electron thermal conductivities, respectively; usually one or the other predominates. The thermal energy associated with phonons or lattice waves is transported in the direction of their motion. The k l contribution results from a net movement of phonons from high to low temperature regions of a body across which a temperature gradient exists. Free or conducting electrons participate in electronic thermal conduction. To the free electrons in a hot region of the specimen is imparted a gain in kinetic energy. They then migrate to colder areas, where some of this kinetic energy is transferred to the atoms themselves (as vibrational energy) as a consequence of collisions with phonons or other imperfections in the crystal. The relative contribution of k e to the total thermal conductivity increases with increasing free electron concentrations, since more electrons are available to participate in this heat transference process. 9

10 Thermal conductivity in Metals, Ceramics and Polymers: Metals In high-purity metals, the electron mechanism of heat transport is much more efficient than the phonon contribution because electrons are not as easily scattered as phonons and have higher velocities. Furthermore, metals are extremely good conductors of heat because relatively large numbers of free electrons exist that participate in thermal conduction. The thermal conductivities of several of the common metals values generally range between about 20 and 400 W/m-K. Since free electrons are responsible for both electrical and thermal conduction in pure metals, theoretical treatments suggest that the two conductivities should be related according to the Wiedemann Franz law L= k σt where σ is the electrical conductivity, T is the absolute temperature, and L is a constant. The theoretical value of L, 2.44 x 10-8 Ω-W/K 2, should be independent of temperature and the same for all metals if the heat energy is transported entirely by free electrons. Alloying metals with impurities results in a reduction in the thermal conductivity, for the same reason that the electrical conductivity is diminished namely, the impurity atoms, especially if in solid solution, act as scattering centers, lowering the efficiency of electron motion. A plot of thermal conductivity versus composition for copper zinc alloys (Figure) displays this effect. Also, stainless steels, which are highly alloyed, become relatively resistive to heat transport. Fig 7: A plot of thermal conductivity versus composition 10

11 Ceramics Nonmetallic materials are thermal insulators inasmuch as they lack large numbers of free electrons. Thus the phonons are primarily responsible for thermal conduction: ke is much smaller than kl. Again, the phonons are not as effective as free electrons in the transport of heat energy as a result of the very efficient phonon scattering by lattice imperfections. Glass and other amorphous ceramics have lower conductivities than crystalline ceramics, since the phonon scattering is much more effective when the atomic structure is highly disordered and irregular. The scattering of lattice vibrations becomes more pronounced with rising temperature; hence, the thermal conductivity of most ceramic materials normally diminishes with increasing temperature, at least at relatively low temperatures. Porosity in ceramic materials may have a dramatic influence on thermal conductivity; increasing the pore volume will, under most circumstances, result in a reduction of the thermal conductivity. In fact, many ceramics that are used for thermal insulation are porous. Heat transfer across pores is ordinarily slow and inefficient. Internal pores normally contain still air, which has an extremely low thermal conductivity approximately 0.02 W/m-K. Furthermore, gaseous convection within the pores is also comparatively ineffective. Polymers Thermal conductivities for most polymers are on the order of 0.3 W/m-K. For these materials, energy transfer is accomplished by the vibration and rotation of the chain molecules. The magnitude of the thermal conductivity depends on the degree of crystallinity; a polymer with a highly crystalline and ordered structure will have a greater conductivity than the equivalent amorphous material. This is due to the more effective coordinated vibration of the molecular chains for the crystalline state. Polymers are often utilized as thermal insulators because of their low thermal conductivities. As with ceramics, their insulative properties may be further enhanced by the introduction of small pores, which are ordinarily introduced by foaming during polymerization. Foamed polystyrene (Styrofoam) is commonly used for drinking cups and insulating chests. Applications of Thermal conductivity: 11

12 There are many applications of thermal conduction in everyday life. Some objects may feel cold to the touch if they are good conductors because they carry away heat from the body rapidly, so a concrete or tiled floor feels much colder to stand on than a carpeted one. A polystyrene cup feels warm to the touch because it conducts away barely any heat from the body. On the other hand, in a very hot room (e.g. Turkish bath), metal objects can feel very hot to the touch and may actually burn the skin. In a block of hot metal the atoms/molecules may vibrate rapidly, perhaps thousands of times each second. If one touches it with one s finger, the rapidly vibrating atoms cause the molecules of the skin to go into sudden and violent motion, resulting in the sensation of pain. Iron-Carbon Phase Diagram: eutectoid Eutectic Pearlite and Cementine Austenite Ferrite Pearlite Pearlite and Carbide Steel Cast iron The above figure shows the iron-carbon phase/equilibrium diagram upto 6.67% C. At 6.67% C, the micro structure of steel is 100% cementite which is represented by the right hand side of the 12

13 boundary. Thus, the iron-carbon equilibrium diagram is the portion between the pure iron and cemenite. Therefore, it is called as iron-iron carbide equilibrium diagram. The cementite hard and brittle, and its melting point is approximately 1560 o C. Pure iron melts at 1539 o C as shown at the left hand side boundary. The line ABCD is the liquidus line and AHJECF is solidus line. Above the liquidus line there is only liquid phase consisting iron and dissolved carbon. Below the solidus line the alloy is completely solid. Region between these two lines represents mixtures of solid and liquid phases. The various critical points marked on the diagram are A 1 (PSK), A 3 (GS) and A cm (SE). It may be noted that PSK is horizontal, which means that lower critical point is same for carbon steels of all compositions. Based on carbon content, iron-iron carbide diagram is divided into two parts, upto two percent carbon it represents steel portion where as above 2% carbon it is cast iron portion. Cast iron portion is further sub dived into hypoeutectic (C< 4.3 %) and hypereutectic (C > 4.3 %). Invariant Reactions: Iron carbon diagram is a complex phase diagram with three invariant reactions. They are peritectic, eutectic and eutectoide reactions. 1. Peritectic reaction: Peritectic invariant reaction occurs at C and composition of 0.18 % of C. The liquid (0.5 % C) combines with δ-iron (0.08 % C) to produce austenite (0.18 % C). This reaction can be written as 2. Eutectic Reaction: Liquid + δ-iron Austenite Eutectic reaction occurs at C and composition of 4.3 % of carbon. The liquid (4.3% C) transforms into eutectic mixture of austenite and cementite. This reaction can be written as Liquid Austenite + Cementite 13 This eutectic is called ledeburite.

14 3. Eutectoid reaction: This is a solid state reaction which occurs at C and composition of 0.8 % C. The austenite (0.8 % C) decomposes into ferrite (0.025 % C) and cementite 6.67 % C. This reaction can be written as Austenite Ferrite+ Cementite This eutectic mixture is called pearlite. Heat treatment of Materials, Hardening, Tempering, Annealing, Normalizing, Quenching, Case-hardening and Solution heat treatment Heat treatment processes for improving mechanical properties of metals Heat treatment is controlled heating and cooling operations used to bring desired change in the physical properties of metals. Its purpose is to improve the structural and physical properties for some particular use or for future work of the metal. There are five basic heat treating processes: hardening, case hardening, annealing, normalizing, and tempering. Each of these processes bring about different results in metal. All the process involves three basic steps: heating, soaking, and cooling. Heating: Heating is the first step in a heat-treating process. Many alloys change structure when they are heated to specific temperatures. The structure of an alloy at room temperature can be a mechanical mixture, a solid solution, or a combination solid solution and mechanical mixture. A mechanical mixture can be compared to concrete. Just as the sand and gravel are visible and held in place by the cement. The elements and compounds in a mechanical mixture are clearly visible and are held together by a matrix of base metal. A solid solution is when two or more metals are absorbed, one into the other, and form a solution. When an alloy is in the form of a solid solution, the elements and compounds forming the metal are absorbed into each other in much the same way that salt is dissolved in a glass of water. The separate elements forming the metal cannot be identified even under a microscope. A metal in the form of a mechanical mixture at room temperature often goes into a solid solution or a partial solution when it is heated. 14

15 Changing the chemical composition in this way brings about certain predictable changes in grain size and structure. Soaking: Once a metal part has been heated to the temperature at which desired changes in its structure will take place, it must remain at that temperature until the entire part has been evenly heated throughout. This is known as soaking. The more mass the part has, the longer it must be soaked. Cooling: After the part has been properly soaked, the third step is to cool it. Here again, the structure may change from one chemical composition to another, it may stay the same, or it may revert to its original form. For example, a metal that is a solid solution after heating may stay the same during cooling, change to a mechanical mixture, or change to a combination of the two, depending on the type of metal and the rate of cooling. All of these changes are predictable. For that reason, many metals can be made to conform to specific structures in order to increase their hardness, toughness, ductility, tensile strength, and so forth. Heat treatment of ferrous metals and non-ferrous metals HEAT TREATMENT OF FERROUS METALS: All heat-treating operations involve the heating and cooling of metals, the common forms of heat treatment for ferrous metals are hardening, tempering, annealing, normalizing, and case hardening. HARDENING: A ferrous metal is normally hardened by heating the metal to the required temperature and then cooling it rapidly by plunging the hot metal into a quenching medium, such as oil, water, or brine. Most steels must be cooled rapidly to harden them. The hardening process increases the hardness and strength of metal, but also increases its brittleness. TEMPERING: Steel is usually harder than necessary and too brittle for practical use after being hardened. Severe internal stresses are set up during the rapid cooling of the metal. Steel is tempered after being hardened to relieve the internal stresses and reduce its brittleness. Tempering consists of heating the metal to a specified temperature and then permitting the metal to cool. The rate of cooling usually has no effect on the metal structure during tempering. Therefore, the metal is usually permitted to cool in still air. Temperatures used for tempering are normally much lower than the hardening temperatures. The higher the tempering temperature used, the softer the metal becomes. High-speed steel is one of the few metals that becomes harder instead of softer after it is tempered. 15

16 ANNEALING: Metals are annealed to relieve internal stresses, soften them, make them more ductile, and refine their grain structures. Metal is annealed by heating it to a prescribed temperature, holding it at that temperature for the required time, and then cooling it back to room temperature. The rate at which metal is cooled from the annealing temperature varies greatly. Steel must be cooled very slowly to produce maximum softness, This can be done by burying the hot part in sand, ashes, or some other substance that does not conduct heat readily (packing), or by shutting off the furnace and allowing the furnace and part to cool together (furnace cooling). NORMALIZING: Ferrous metals are normalized to relieve the internal stresses produced by machining, forging, or welding. Normalized steels are harder and stronger than annealed steels. Steel is much tougher in the normalized condition than in any other condition. Parts that will be subjected to impact and parts that require maximum toughness and resistance to external stresses are usually normalized. Normalizing prior to hardening is beneficial in obtaining the desired hardness, provided the hardening operation is performed correctly. Low carbon steels do not usually require normalizing, but no harmful effects result if these steels are normalized. Normalizing is achieved by heating the metal to a specified temperature (which is higher than either the hardening or annealing temperatures), soaking the metal until it is uniformly heated, and cooling it in still air. CASE HARDENING: Case hardening is an ideal heat treatment for parts which require a wearresistant surface and a tough core, such as gears, cams, cylinder sleeves, and so forth. The most common case-hardening processes are carburizing and nitriding. During the case-hardening process, a low-carbon steel (either straight carbon steel or low-carbon alloy steel) is heated to a Specific temperature in the presence of a material (solid, liquid, or gas) which decomposes and deposits more carbon into the surface of a steel. Then, when the part is cooled rapidly, the outer surface or case becomes hard, leaving the, inside of the piece soft but very tough. HEAT TREATMENT OF NONFERROUS METALS: Two types of heat-treating operations can be performed on nonferrous metals. They are annealing and solution heat treating. ANNEALING: Most nonferrous metals can be annealed. The annealing process consists of heating the metal to a specific temperature, soaking, and cooling to room temperature. The 16

17 temperature and method of cooling depend on the type of metal. Annealing is often accomplished after various cold working operations because many nonferrous metals become hard and brittle after cold working. Also, annealing is used to remove the effects of solution heat treatment so that machining or working qualities can be improved. SOLUTION HEAT TREATMENT: The tensile strength of many nonferrous alloys can be increased by causing the materials within the alloy to go into a solid solution and then controlling the rate and extent of return to an altered mechanical mixture. This operation is called solution heat treatment. After an alloy has been heated to a specified temperature, it is quenched or cooled rapidly, which traps the materials in the solid solution attained during the heating process. From this point, the process varies greatly depending on the metal. To be sure the materials in the alloy do not revert to their original configuration after a period of time, a process of aging or precipitation hardening must follow. In this process the materials in the alloy are allowed to change or to precipitate out of the solid solution. This process occurs under controlled conditions so that the resultant grain structure will produce a greater tensile strength in the metal than in its original condition. Depending on the alloy, this precipitation process can also consist of simply aging the alloy at room temperature for a specified time and then air-cooling it; this is called artificial aging. Aluminum alloys can be obtained in various conditions Problems 1. Estimate energy required to raise the temperature of 5 kg of materials, Aluminium and Brass from 20 to 150 C. Cp of Aluminium and Brass are 900 J/kg-K and 375 J/kg-K respectively. Comment on the result. 2. A 0.4 m long rod of copper elongates 0.48 mm on heating from 20 to 100 C. Find linear coefficient of thermal expansion of the copper rod. A copper wire 15 m long is cooled from 40 to -9 C. Find change in length it experienced. 17

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