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https://docs.google.com/document/d/1SxCwaMe40jDQmCeWcBUP61t1NDUi1gr7/edit?usp=sharing&ouid=109474854956598892099&rtpof=true&sd=tru cooling Heat transfer can take place from one place to the other by three different processes namely conduction, convection and radiation. HEAT CONDUCTION Conduction usually takes place in solids. Steady State When heat conduction takes place across say a rod of certain material, the state at which each cross-section of rod is at a constant temperature (which is different for different sections) is called steady state. The bar does not absorb any heat, and if the rod is completely lagged then the heat entering one end is equal to the heat leaving other end. Law of Conduction l T1 ⇒ A T2 x dx Rate of heat flow across any section is given by dQ = kA dT CHAPTER COVERS : Heat Conduction Steady State Thermal Resistance Series and Parallel Rods Formation of Ice Layer Convection Radiation Kirchhoff's Law Stefan's Law Newton's Law of Cooling Wein's Displacement Law dt dx dT Here k = Thermal conductivity and dx is known as temperature gradient i.e. rate of change of temperature with distance. Units of thermal conductivity are Watt (metre-kelvin)–1 or Wm–1K–1 Important Results In steady state, rate of heat flow is same across any section dQ = kA ⎛ T1 − T2 ⎞ ⎜ ⎟ dt ⎝ l ⎠ TEACHING CARE Online Live Classes https://www.teachingcare.com/ +91-9811000616 If temperature at every cross section remains constant, temperature at a distance x from T1 end is T = T ⎛ T1 − T2 ⎞ x Tx x 1 ⎜ ⎟ ⎝ l ⎠ 1 T2 x l Graphical Variation of Tx with x Lagged rod T1 x dx T2 x Unlagged rod T1 T2 x Decreasing Order of Conductivity For some special cases it is as follows : KAg > KCu > KAl Ksolid > Kliquid > Kgas Kmetals > Knon-metals THERMAL RESISTANCE OF A ROD In steady state dQ = kA (T1 − T2 ) dt l T1 T2 R = l ⎡as in current electricity R = ρl = l ⎤ A kA ⎢⎣ A σA ⎥⎦ l Weidman – Frenz law k = constant σT Where σ is electrical conductivity Composite Rod : Series In steady state l l l + l R = 1 + 2 = 1 2 A A k1A A kA 1 T2 Where k = effective thermal conductivity given by l1 T l2 k = l1 + l 2 l1 + l2 k1 k2 k1 T k2 T Temperature of junction T = l1 1 k l2 . For same geometrical dimensions, k 1 + 2 T = k1T1 + k2T2 k1 + k2 In parallel dQ = dQ1 + dQ2 l1 l2 l dt dt 1 = 1 + dt 1 ⇒ k(A1 + A2 ) = k1A1 + k2 A2 R R1 R2 l l l where k = effective coefficient of thermal conductivity given by k = k1A1 + k2 A2 A1 + A2 Applications T1 T2 A dQ1 1 k1 dt A k dQ2 2 2 dt If number of conductors having identical dimension are in series then equivalent thermal conductivity is harmonic mean of individuals, n Keq = 1 + 1 K1 K2 + .......... . + 1 Kn If number of conductors having identical dimension are in parallel then equivalent thermal conductivity is arithmatic mean of individuals, Keq = K1 + K2 + K3 + ... + Kn n 2 k1A1 l T1 + k2 A2 l T2 + k3 A3 T T1 3 (3) T = 1 2 3 k1A1 l1 k2 A2 l 2 k3 A3 l3 3 T = k1T1 + k2T2 + k3T3 k1 + k2 + k3 kA = k1A1 + k2A2 If R1 = R & R2 = 2R, then [l1 = l2 = l3, A1 k = k1 + 3k2 4 = A2 = A3] Formation of ice If temperature of water just below the ice layer = 0°C –θ °C dx water 0°C ⇒ Rate of formation ∝ 1 instantane ous thickness ρL x 2 Time taken to deposit x thickness, t = kθ 2 ⇒ t ∝ x2 Let there be a hollow sphere of inner radius r1 and outer radius r2, having inner surface temperature T1 and outer surface temperature T2, To calculate radial rate of flow of heat 'H', assume a spherical shell of radius r and width dr. The heat flow rate through this section is given by H = 4πKr1r2 (T1 − T2 ) r2 − r1 H ∝ r1r2 r2 − r1 CONVECTION In this process, heat is transferred from one place to the other by the actual movement of heated substance. To understand the convection process consider a beaker in which some liquid is placed as shown in the figure. When liquid is heated at the bottom, the liquid expands and hence pressure at that point (A) reduces. So liquid from B & C D F E moves toward point A. Now to take position of liquid at B & C liquid from D & Burner E moves towards B & C respectively. At D & E liquid is supplied from region F and to fill the vacant place F liquid moves from A to F. In this way convention currents set up on the whole beaker. Natural Convection : This type of convection results from difference in densities due to difference in temperature. Forced Convection : Here, heated fluid is forced to move by a blower. RADIATION The process by which heat is transferred directly from one body to another, without requiring any medium is called radiation. Radiation is the fastest mode of heat transfer as in this mode, heat energy is propagated at speed of light, i.e., 3 × 108 m/s. As all bodies radiate energy at all temperatures (more than θK) and at all times, radiation from a body can never be stopped. Heat radiations are invisible, travel in straight line, cast shadow, affect photographic plate and can be reflected by mirrors and refracted by glasses. Blue star is hotter than red star. A medium which allows heat radiations to pass through it without absorbing them is called diathermanous medium. e.g. dry air. A medium which partly absorbs heat rays is called athermanous medium. e.g. moist air, metals, wood, glass etc. Glass and water vapours transmit shorter wavelengths through them but reflects longer wavelengths. This concept is utilised in Green House Effect. Glass transmits those waves which are emitted by a source at a temperature greater than 100ºC. So heat rays emitted from sun are able to enter through glass enclosure but heat emitted by small plants growing in the nursery gets trapped inside the enclosure. Perfectly Black Body A body which absorbs all the radiations incident on it is called perfectly black body. Absorptive Power (a) : Absorptive power of a surface is the ratio of the radiant energy absorbed by it in a given time interval to total radiant energy incident on it in the same time interval. Absorptive power of a black body is maximum i.e. unity. Emissive Power (e) : Emissive power of a surface is defined as the radiant energy emitted per second per unit area of the surface. Spectral Emissive Power : The radiant energy/second/area corresponding to a definite wavelength is called spectral emissive power. If eλ is spectral emissive power and e is emissive power then ∞ e = ∫eλdλ 0 Emissive power of a surface depends on its nature and temperature. Its units are W/m2. KIRCHHOFF’S LAW The ratio of emissive power to absorptive power for a given wavelength is same for all surfaces at the same temperature, and is equal to the emissive power of a perfectly black body for that wavelength at that temperature. This implies that a good absorber is a good emitter. Following points must be remembered. Sand is rough and black. Therefore it is a good absorber as well as good emitter. A polished metal plate has a black spot. When the plate is heated strongly and taken to a dark room, spot will appear brighter than the plate. In sodium absorption spectrum, two dark lines in yellow region are found. If emission spectrum of sodium is observed, it is found to emit the corresponding lines. Fraunhoffer lines are dark lines in spectrum of sun and are formed because, the elements present in outer atmosphere absorb their characteristic wavelengths. STEFAN’S LAW The radiant energy emitted by a perfectly black body per second per unit area (emissive power) is directly proportional to the fourth power of the absolute temperature of the body. R ∝ T 4 R = σT 4 R = Power Area ⇒ P = AσT 4 For other bodies P = εAσT4, ε is emissivity of the body. Rate of heat loss For a sphere of radius r at a temperature T placed in a surrounding of temperature T0, the rate of heat loss is dQ = 4πr 2εσ(T 4 − T 4 ) , Where ε is emissivity. dt 0 Rate of cooling It is rate of fall in temperature, It is given by dT 3εσ(T 4 − T 4 ) = − 0 dt ρsr Newton’s Law of Cooling If the temperature T of a body is not much different from surrounding temperature T0, then rate of cooling of a liquid is directly proportional to the difference in the temperature of liquid T and temperature of surroundings i.e. Rate of cooling − dT dt ∝ (T − T0 ) Results Tf = T0 + (Ti – T0)e–αt, where Ti is initial temperature, Tf is temperature after time t. Another form αt = log Ti − T0 Tf − T0 Ti ⎡ ⎤ (3) dT dt = 4εAσT 3 mc (T − T0 ⎢m = mass of body⎥ ), ⎢ c = specific heat ⎥ T0 A = surface area ⎢ ⎥ ⎢ ε = emissivity ⎥ t Another approximate formula is T1 − T2 = α⎛ T1 + T2 − T ⎞ ⎜ 0 ⎟ t ⎝ 2 ⎠ Above formula gives time ‘t’ taken by the body to cool from T1 to T2. T0 is temperature of surrounding. Thermometric Conductivity or Diffensivity It is defined of the ratio of thermal conductivity to the thermal capacity per unit volume of material. D = K ρc K → thermal conductivity ρ → density c → specific heat Practical examples : Hot water loses heat in smaller duration as compared to moderate warm water. Adding milk in tea reduces the rate of cooling. WEIN’S DISPLACEMENT LAW This law states that the wavelength corresponding to maximum intensity for a black body is inversely proportional to the absolute temperature of the body λ = b m T where b is a constant known as Wein's constant Results (1) λmax T = b (2) b = 2.898 × 10–3 m-K Area under eλ – λ graph = σT 4 If the temperature of the black body is made two fold, λmax becomes half, while area becomes 16 times. Temperature of the Sun, If T = temperature of sun, then total energy radiated by sun per second = ρT4 (4πR2) Intensity at distance r from the sun (i.e., on earth) σT 4R 2 2 I = S (Solar constant) = r 2 [S = 1.4 KW / m ] ⎡⎛ r 2 ⎞ S ⎤1 4 ⎢⎜ R 2 ⎟ σ ⎥ ⎣⎝ ⎠ ⎦ ❑ ❑ ❑