CONDUCTION Conduction The transfer of energy from the

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CONDUCTION Conduction: The transfer of energy from the more energetic particles of a substance

CONDUCTION Conduction: The transfer of energy from the more energetic particles of a substance to the adjacent less energetic ones as a result of interactions between the particles. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. The rate of heat conduction through a plane layer is proportional to the temperature difference across the layer and the heat transfer area, but is inversely proportional to the thickness of the layer. Heat conduction through a large plane wall of thickness x and area A.

When x → 0 Fourier’s law of heat conduction Thermal conductivity, k: A measure

When x → 0 Fourier’s law of heat conduction Thermal conductivity, k: A measure of the ability of a material to conduct heat. Temperature gradient d. T/dx: The slope of the temperature curve on a T-x diagram. Heat is conducted in the direction of decreasing temperature, and the temperature gradient becomes negative when temperature decreases with increasing x. The negative sign in the equation ensures that heat transfer in the positive x direction is a positive quantity. In heat conduction analysis, A represents the area normal to the direction of heat transfer. The rate of heat conduction through a solid is directly proportional to its thermal conductivity.

Thermal Conductivity Thermal conductivity may be defined as the amount of heat conducted per

Thermal Conductivity Thermal conductivity may be defined as the amount of heat conducted per unit time across unit area and through unit thickness when a temperature difference of unit degree is maintained across the bounding surfaces. The thermal conductivity of a material is a measure of the ability of the material to conduct heat. A high value for thermal conductivity indicates that the material is a good heat conductor, and a low value indicates that the material is a poor heat conductor or insulator. A simple experimental setup to determine thermal conductivity of a material.

The range of variation of thermal conductivity of different classes of materials at room

The range of variation of thermal conductivity of different classes of materials at room temperature

The range of thermal conductivity of various materials

The range of thermal conductivity of various materials

Thermal Conductivity Thermal Conductivities (average values at normal pressure and temperature) of some common

Thermal Conductivity Thermal Conductivities (average values at normal pressure and temperature) of some common materials are as under Material k (W/m K) Diamond 2300 Brick 0. 72 Silver 429 Water (l) 0. 613 Copper 401 Wood (oak) 0. 17 Gold 317 Helium (g) 0. 152 Aluminium 237 Refrigerant 12 0. 072 Iron 80. 2 Glass fibre 0. 043 Mercury (l) 8. 54 Air (g) 0. 026 Glass 0. 78

Thermal Conductivity The thermal conductivity is a property of material which depends upon •

Thermal Conductivity The thermal conductivity is a property of material which depends upon • Material structure (chemical physical state and texture) • Density of the material • Moisture content • Pressure and temperature composition,

Thermal Conductivity The thermal conductivity of a material is due to flow of free

Thermal Conductivity The thermal conductivity of a material is due to flow of free electrons (in case of metals) and lattice vibrations waves (in case of fluids) The mechanisms of heat conduction in different phases of a substance.

Thermal Conductivity Accordingly thermal conductivity of a material is the outcome of migration of

Thermal Conductivity Accordingly thermal conductivity of a material is the outcome of migration of free electrons and lattice vibrational waves. In metal molecules are closely packed; molecular activity is rather small and so thermal conductivity is substantially due to the flow of free electrons. In fluids, the free electron movement is negligible small and therefore thermal conductivity results primarily from the frequency of interactions between the lattice atoms.

Thermal Conductivity Further metals are the best conductors while liquids are generally poor conductors.

Thermal Conductivity Further metals are the best conductors while liquids are generally poor conductors. Probably the disordered structure of the liquids and so of the gases is not conductive for transmitting molecular vibration.

Thermal Conductivity of Solids Thermal conductivity of solids is made up of two components

Thermal Conductivity of Solids Thermal conductivity of solids is made up of two components 1. Due to flow of free electrons and 2. Due to lattice vibration (atom which are bound in a periodic arrangement called lattice) First effect is known as electronic conduction and second effect is known as photon conduction

Metals and alloys In case of pure metals and alloys a) There is an

Metals and alloys In case of pure metals and alloys a) There is an abundance of free electrons and the electronic conduction predominates. Since free electrons are also responsible for electrical conduction, it is observed that good electrical conductors are also good thermal conductors e. g. copper, silver etc.

Metals and alloys In case of pure metals and alloys a) There is an

Metals and alloys In case of pure metals and alloys a) There is an abundance of free electrons and the electronic conduction predominates. Since free electrons are also responsible for electrical conduction, it is observed that good electrical conductors are also good thermal conductors e. g. copper, silver etc.

Metals and alloys b) Any effect which inhibits the flow of free electrons in

Metals and alloys b) Any effect which inhibits the flow of free electrons in pure metals reduces the value of thermal conductivity. For example with a rise in temperature, the lattice vibration increases and this offers a resistance to the flow of electrons and therefore for pure metals thermal conductivity decreases as temperature increases (aluminum and uranium being the exception) Thermal conductivities of materials vary with temperature

Metals and alloys Thermal conductivity of aluminum stays almost constant within temperature range of

Metals and alloys Thermal conductivity of aluminum stays almost constant within temperature range of 130 0 C to 370 0 C. Most of the outer electrons of the uranium atoms are tied up in covalent bonds and as such the contribution of free electrons to conduction process is small. Conduction of heat within uranium depends mainly on the vibration of atoms. The vibration tendency increases with temperature rise and so does thermal conductivity of uranium.

Metals and alloys Variation of thermal conductivity with temperature for a few metals

Metals and alloys Variation of thermal conductivity with temperature for a few metals

Metals and alloys c) Alloying decreases the value of thermal conductivity since the foreign

Metals and alloys c) Alloying decreases the value of thermal conductivity since the foreign atoms cause scattering of free electrons, thus impending their free flow through the material. Pure metals have very high thermal conductivity. Impurities or alloying element reduce thermal conductivity considerably.

Metals and alloys Thermal conductivity of pure copper near about room temperature is 401

Metals and alloys Thermal conductivity of pure copper near about room temperature is 401 W/m 0 C while presence of traces of arsenic reduces the value of thermal conductivity to 142 W/m 0 C.

Metals and alloys d) Thermal conductivity of a metal varies considerably when it (metal)

Metals and alloys d) Thermal conductivity of a metal varies considerably when it (metal) is heat treated or mechanically processed/formed (forging, drawing and bending). • Heat treatment and mechanical forming reduce the value of thermal conductivity of pure metals. • For example, thermal conductivity of hardened steel is lower than that of annealed state.

Metals and alloys e) Thermal conductivity of alloys generally increases as temperature increases. Variation

Metals and alloys e) Thermal conductivity of alloys generally increases as temperature increases. Variation of thermal conductivity with temperature for a few alloys

Metals and alloys f) Since the phenomenon of electron conduction is responsible for both

Metals and alloys f) Since the phenomenon of electron conduction is responsible for both thermal conduction and electrical conduction, it is reasonable to presume that there must be relation between these two quantities. In fact, Weidemann-Franz law gives this relation.

Metals and alloys The Wiedemann and Franz law (based on experimental results) regarding thermal

Metals and alloys The Wiedemann and Franz law (based on experimental results) regarding thermal and electrical conductivities of a material states as follows: “The ratio of thermal and electrical conductivities is the same for all metals at the same temperature and that the ratio is directly proportional to the absolute temperature of the metal”.

Metals and alloys Mathematically k/σ α T k/σT = C Where k =Thermal conductivity

Metals and alloys Mathematically k/σ α T k/σT = C Where k =Thermal conductivity of metal at temperature T (K) (W/ m K) σ = Electrical conductivity of metal at temperature T (K) (ohm m)-1 and C = Constant (for all metals) referred to as Lorenz number (2. 45× 10 -8 W Ohms/K 2)

Metals and alloys • An important application of Wiedemann and Franz law is to

Metals and alloys • An important application of Wiedemann and Franz law is to determine the value of thermal conductivity of a metal at a desired temperature, knowing the value of electrical conductivity at the same temperature. Note that it is easier to measure experimentally the value of electrical conductivity than that of thermal conductivity. • Wiedemann and Franz law conveys that the materials which are good conductors of electricity are also conductors of heat.

Non-metallic Solids a) Non-metallic solids do not conduct heat (there are no free electrons)

Non-metallic Solids a) Non-metallic solids do not conduct heat (there are no free electrons) as efficiently as metals and hence thermal conductivity values are much lower than those of metals. For many of the building and insulating materials (concrete, stone, brick, glass wool, cork etc. ) thermal conductivity may vary from sample to sample due to variations in structure, composition, density and porosity.

Non-metallic Solids For heat insulating materials, general range of values of k are from

Non-metallic Solids For heat insulating materials, general range of values of k are from 0. 023 W/m 0 C to 2. 9 W/m 0 C. Thermal conductivity increases with temperature for insulating materials. Variation of thermal conductivity with temperature for insulating materials

Non-metallic Solids b) For porous heat insulating material (brick, concrete, asbestos, slag etc. ),

Non-metallic Solids b) For porous heat insulating material (brick, concrete, asbestos, slag etc. ), thermal conductivity depends greatly on density of the material and the type of gas or liquid filling the voids. Presence of air filled pores and cavities reduce thermal conductivity because then the heat has to be transferred across many air spaces and air is known to be poor heat conductor.

Non-metallic Solids Thermal conductivity of porous materials also depends on the moisture content in

Non-metallic Solids Thermal conductivity of porous materials also depends on the moisture content in the material; k of a damp material is much higher than that of the dry material and water taken individually. For dry brick k = 0. 35 W/m-deg For water k = 0. 60 W/m-deg For damp brick k = 1 W/m-deg This behavior is may be attributed to (i) capillary movement of water with in the pores which results in convection heat transfer (ii) properties of the absorbed moisture are different from those of free moisture.

Non-metallic Solids Density is another parameter that affects thermal conductivity of material; thermal conductivity

Non-metallic Solids Density is another parameter that affects thermal conductivity of material; thermal conductivity increases with density growth. For example, k of asbestos increases from 0. 105 to 0. 248 W/m 0 C as density increases from 400 to 8000 kg/m 3

Non-metallic Solids c) Thermal conductivity of granular materials increases with temperature since with increasing

Non-metallic Solids c) Thermal conductivity of granular materials increases with temperature since with increasing temperature, radiation from the granules also comes into picture along with conduction of medium filling the spaces.

Non-metallic Solids Materials having a crystalline structure have a high value of thermal conductivity

Non-metallic Solids Materials having a crystalline structure have a high value of thermal conductivity than the substances in amorphous form. For quartz (a solid with crystalline structure) k = 30. 5 W/m-deg at -100 0 C = 10. 4 W/m-deg at +100 0 C For pyrex (a substance of amorphous form) k = 1. 02 W/m-deg at 0 0 C = 1. 73 W/m-deg at 500 0 C Irregular arrangement of the atoms in case of amorphous solids inhibits the effectiveness of heat transfer by molecular impact.

Variation of thermal conductivity of solids with temperature: In heat transfer calculations, generally we

Variation of thermal conductivity of solids with temperature: In heat transfer calculations, generally we assume k to be constant when the temperature range is small; however the temperature range if large, it is necessary to take into account the variation of k with temperature.

Usually, for solids, a linear variation of thermal conductivity with temperature can be assumed

Usually, for solids, a linear variation of thermal conductivity with temperature can be assumed without loss of much accuracy. k(T) = k 0 ( 1+βT) Where k(T) = thermal conductivity at desired temperature T, W/m 0 C k 0= thermal conductivity at reference temperature at 0 0 C, W/m 0 C β = a temperature coefficient, 1/ 0 C T = temperature, 0 C

Representative values of k 0 and β

Representative values of k 0 and β

Variation of thermal conductivity with temperature for a few pure metals It may be

Variation of thermal conductivity with temperature for a few pure metals It may be noted that the variation is linear.

Value of β may be positive or negative. Generally β is negative for metals

Value of β may be positive or negative. Generally β is negative for metals (exception being aluminum, uranium and certain nonferrous alloys) and positive for non metals and insulators (magnesite bricks being exception) and alloys.

Thermal Conductivity of Liquids Non-metallic liquids Heat propagation in liquid is considered to be

Thermal Conductivity of Liquids Non-metallic liquids Heat propagation in liquid is considered to be due to elastic oscillations. As per this hypothesis, thermal conductivity of liquids is given by Where cp = specific heat of liquid at constant pressure ρ = density of liquid M = Molecular weight of liquid A = Constant depending on the velocity of elastic wave propagation in the liquid; it does not depend on nature of liquid; but on temperature

Non-metallic liquids It is noted that the product A. cp is nearly constant. As

Non-metallic liquids It is noted that the product A. cp is nearly constant. As temperature rises, density of liquid falls and as per equation The value of thermal conductivity also drops for liquids with constant molecular weights (i. e. for nonassociated or slightly associated liquids) Notable exceptions are water and glycerin, which are heavily associated liquids. With rising pressure, thermal conductivity of liquids increases. For liquid k value ranges from 0. 07 to 0. 7 W/m 0 C

Non-metallic liquids Thermal conductivity of non-metallic liquids

Non-metallic liquids Thermal conductivity of non-metallic liquids

Liquid metals like sodium, potassium etc. are used in high flux applications as in

Liquid metals like sodium, potassium etc. are used in high flux applications as in nuclear power plants where a large amount of heat has to be removed in small area. Thermal conductivity values of liquid metals are much higher than those for non-metallic liquids. For example, liquid sodium at 644 K has k=72. 3 W/ m K; liquid potassium at 700 K has k=39. 5 W/m K; and liquid bismuth at 589 K has k=16. 4 W/ m K

Thermal Conductivity of Gases a) Heat transfer by conduction in gases at ordinary pressure

Thermal Conductivity of Gases a) Heat transfer by conduction in gases at ordinary pressure and temperature is explained by the Kinetic Theory of gases. Temperature is a measure of kinetic energy of molecules. Radom movement and collision of gas molecules contribute to the transport of kinetic energy, and therefore to transport of heat. So, the two quantities that come into picture now are: the mean molecular velocity V and the mean free path, I. Mean free path is defined as the mean distance travelled by a molecule before it collides with.

Thermal Conductivity of Gases Thermal conductivity of gases is given by Where V =

Thermal Conductivity of Gases Thermal conductivity of gases is given by Where V = mean molecular velocity l = mean free path (average distance travelled by a molecule before experiencing collision) cv = specific heat of gas at constant volume ρ = density

Thermal Conductivity of Gases b) As pressure increases, density ρ increases, but the mean

Thermal Conductivity of Gases b) As pressure increases, density ρ increases, but the mean free path l decreases almost by the same proportion and the product l ρ remains almost constant, i. e. thermal conductivity of gases does not vary much with pressure except at very low (less than 20 mm. Hg) or very high (more than 20, 000 bar) pressures. c) As to the effect of temperature on thermal conductivity of gases, mean molecular velocity V depends on temperature as follows, Where G = Universal gas constant = 8314. 2 J/kmol K M = Molecular weight of gas; T = absolute temperature of Gas K

Thermal Conductivity of Gases i. e. mean molecular velocity varies directly as the square

Thermal Conductivity of Gases i. e. mean molecular velocity varies directly as the square root of absolute temperature and inversely as the square root of the molecular weight of a gas. Specific heat cv also increases as temperature increases. As a result, thermal conductivity of gases increases as temperature increases.

Thermal Conductivity of Gases d) For the reason stated above, gases with higher molecular

Thermal Conductivity of Gases d) For the reason stated above, gases with higher molecular weight have small thermal conductivity than those with lower molecular weight. For example k for hydrogen (mol wt = 2) = 0. 190 W/ m-deg k for oxygen (mol wt = 32) = 0. 0272 W/ m-deg

Thermal Conductivity of Gases f) Generally, thermal conductivity values for gases vary in the

Thermal Conductivity of Gases f) Generally, thermal conductivity values for gases vary in the range of 0. 006 to 0. 6 W/m 0 C g) Thermal conductivity of steam and other imperfect gases depend very much on pressure unlike that of perfect gases.

Thermal Conductivity of Gases Variation of k with temperature for a few gases Variation

Thermal Conductivity of Gases Variation of k with temperature for a few gases Variation of k with temperature for hydrogen and helium

Insulation systems Materials with large thermal conductivity are called thermal conductors and those with

Insulation systems Materials with large thermal conductivity are called thermal conductors and those with small thermal conductivity are called thermal insulators. Insulating materials are used for obstructing the flow of heat between an enclosure and its surroundings. Insulation is required for high temperature systems as well as low temperature systems. Insulation systems may be classified as (i) Fibrous (ii) Cellular (iii) Powder (iv) Reflective

Insulation systems In high temperature systems, any leakage of heat from boilers, furnaces or

Insulation systems In high temperature systems, any leakage of heat from boilers, furnaces or piping carrying hot fluids represents an energy loss. Similarly in low temperature/cryogenic systems, any heat leakage into the low temperature region represents an energy loss.

Insulation systems Low temperature insulation (cork, rock, wool, glass wool, cattle hair, slag wool

Insulation systems Low temperature insulation (cork, rock, wool, glass wool, cattle hair, slag wool and thermocole etc. ) are used when the enclosure is at a temperature lower than the ambient temperature and it is desired to prevent the enclosure from gaining heat. High temperature insulations (asbestos, diatomaceous earth, magnesia etc. ) are used when it is desired to prevent an enclosure at a temperature higher than the ambient from losing heat to surroundings.

Insulation systems Super insulators include powders, fibres or multilayer materials that have been evacuated

Insulation systems Super insulators include powders, fibres or multilayer materials that have been evacuated of all air. The low conductivity of insulating materials is due primarily to air (a poorly conducting gas) that is contained in the pores rather than the low conductivity of the solid substance. Substances under low temperature conditions that have exceeding high thermal conductivity are known as super conductors. For example thermal conductivity of aluminium reaches a value of 20000 W/ m-deg at 10 K and this is over 100 times as large as the value that occurs at room temperature.

Insulation systems Common Insulation used in Industry

Insulation systems Common Insulation used in Industry

Thermal Diffusivity cp Specific heat, J/kg · °C: Heat capacity per unit mass cp

Thermal Diffusivity cp Specific heat, J/kg · °C: Heat capacity per unit mass cp Heat capacity, J/m 3·°C: Heat capacity per unit volume Thermal diffusivity, m 2/s: Represents how fast heat diffuses through a material A material that has a high thermal conductivity or a low heat capacity will obviously have a large thermal diffusivity. The larger thermal diffusivity, the faster the propagation of heat into the medium. A small value of thermal diffusivity means that heat is mostly absorbed by the material and a small amount of heat is conducted further.

Topics • General Heat Conduction equation in Cartesian Coordinates • General Heat Conduction Cylindrical

Topics • General Heat Conduction equation in Cartesian Coordinates • General Heat Conduction Cylindrical Coordinates • General Heat Conduction equation in Spherical Coordinates equation in

Topics Heat Conduction through a plane wall and composite walls • Heat Conduction through

Topics Heat Conduction through a plane wall and composite walls • Heat Conduction through a plane wall § Case I: Uniform k § Case II: Variable k • Heat Conduction through a composite walls • The overall heat transfer coefficient

Topics Heat Conduction through Hollow and composite cylinder • Heat Conduction through a hollow

Topics Heat Conduction through Hollow and composite cylinder • Heat Conduction through a hollow cylinder § Case I: Uniform k § Case II: Variable k • Heat Conduction through a composite cylinder

Topics Heat Conduction through Hollow and composite sphere • Heat Conduction through a hollow

Topics Heat Conduction through Hollow and composite sphere • Heat Conduction through a hollow sphere § Case I: Uniform k § Case II: Variable k • Heat Conduction through a composite sphere

Topics Critical thickness of Insulation • Insulation – General Aspects • Critical thickness of

Topics Critical thickness of Insulation • Insulation – General Aspects • Critical thickness of insulation Heat Transfer from extended surfaces (Fins) • Introduction • Heat flow through rectangular fin § Heat dissipation from an infinitely long fin § Heat dissipation from a fin insulated at the tip § Heat dissipation from a fin losing heat at the tip § Efficiency and effectiveness of fin