Chapter 13: Transmission of Heat PDF

60 128 Transmission of Heat 14.1 Introduction. E3 Heat energy transfers from a body at higher temperature to a body at lower temperature. The transfer of heat from one body to another may take place by one of the following modes. Convection Radiation Heat flows from hot end to cold end. Particles of...

60 128 Transmission of Heat 14.1 Introduction. E3 Heat energy transfers from a body at higher temperature to a body at lower temperature. The transfer of heat from one body to another may take place by one of the following modes. Convection Radiation Heat flows from hot end to cold end. Particles of the medium simply oscillate but do not leave their place. Each particle absorbing heat is mobile Heat flows without any intervening medium in the form of electromagnetic waves. Medium is necessary for conduction Medium is convection Medium is not necessary for radiation It is a slow process It is also a slow process ID Conduction D YG U necessary for It is a very fast process Path may be zig-zag or curved Path is a straight line Conduction takes place in solids Convection takes place in fluids Radiation takes place in gaseous and transparent media The temperature of the medium increases through which heat flows In this process also the temperature of medium increases There is no change in the temperature of the medium U Path of heat flow may be zigzag ST 14.2 Conduction. The process of transmission of heat energy in which the heat is transferred from one particle to other particle without dislocation of the particle from their equilibrium position is called conduction. (i) Conduction is a process which is possible in all states of matter. (ii) In solids only conduction takes place. (iii) In non-metallic solids and fluids the conduction takes place only due to vibrations of molecules, therefore they are poor conductors. Transmission of Heat 129 (iv) In metallic solids free electrons carry the heat energy, therefore they are good conductor of heat. (1) Variable and steady state 60 When one end of a metallic rod is heated, heat flows by conduction from the hot end to the cold end. In the process of conduction each cross-section of the rod receives heat from the adjacent E3 cross-section towards the hot end. A part of this heat is absorbed by the cross-section itself whose temperature increases, another part is lost into atmosphere by convection & radiation and the rest is conducted away to the next cross-section. Because in this state temperature of every cross-section of the rod goes on increasing, Temperatur e ID hence rod is said to exist in variable state. Cold end Metallic rod D YG Distance from hotter end U Hot end After sometime, a state is reached when the temperature of every cross-section of the rod becomes constant. In this state, no heat is absorbed by the rod. The heat that reaches any crosssection is transmitted to the next except that a small part of heat is lost to surrounding from the sides by convection & radiation. This state of the rod in which no part of rod absorbs heat is called steady state. U (2) Isothermal surface ST Any surface (within a conductor) having its all points at the same temperature, is called isothermal surface. The direction of flow of heat through a conductor at any point is perpendicular to the isothermal surface passing through that point. (i) If the material is rectangular or cylindrical rod, the isothermal surface is a plane surface. (ii) If a point source of heat is situated at the centre of a sphere the isothermal surface will be spherical, (iii) If steam passes along the axis of the hollow cylinder, heat will flow through the walls of the cylinder so that in this condition the isothermal surface will be cylindrical. Q Q Q Q S Cylindrical isothermal surface Plane isothermal surfaces Spherical isothermal surface 130 Transmission of Heat 60 (3) Temperature Gradient The rate of change of temperature with distance between two isothermal surfaces is called temperature gradient. E3 If the temperature of two isothermal surfaces be  and (    ) , and the perpendicular distance between them be x then Temperature gradient = (   )    = x x  ( –  ) Q ID The negative sign show that temperature  decreases as x the distance x increases in the direction of heat flow. Unit : K/m (S.I.) and Dimensions : [L1 ] U (4) Coefficient of thermal conductivity If L be the length of the rod, A the area of cross-section and 1 and 2 are the temperature of D YG its two faces, then the amount of heat flowing from one face to the other face in time t is given by Q KA( 1   2 ) t l Where K is coefficient of thermal conductivity of material of A 2 Q l U rod. It is the measure of the ability of a substance to conduct heat through it. 1 If A = 1m2, (1 – 2) = 1oC, t = 1 sec and l = 1m, then Q = K. ST Thus, thermal conductivity of a material is the amount of heat flowing per second during steady state through its rod of length 1 m and cross-section 1 m2 with a unit temperature difference between the opposite faces. (i) Units : Cal/cm-sec oC (in C.G.S.), kcal/m-sec-K (in M.K.S.) and W/m- K (in S.I.) (ii) Dimension : [MLT 3 1 ] (iii) The magnitude of K depends only on nature of the material. (iv) For perfect conductors, K   and for perfect insulators, K  0 Transmission of Heat 131 (v) Substances in which heat flows quickly and easily are known as good conductor of heat. They possesses large thermal conductivity due to large number of free electrons. Example : Silver, brass etc. 60 (vi) Substances which do not permit easy flow of heat are called bad conductors. They possess low thermal conductivity due to very few free electrons. Example : Glass, wood etc. (vii) The thermal conductivity of pure metals decreases with rise in temperature but for alloys thermal conductivity increases with increase of temperature. (5) Applications of conductivity in daily life E3 (viii) Human body is a bad conductor of heat (but it is a good conductor of electricity). (i) Cooking utensils are provided with wooden handles, ID because wood is a poor conductor of heat. The hot utensils can be easily handled from the wooden handles and our hands are saved from burning. Frying pan Wooden handle U (ii) We feel warmer in a fur coat. The air enclosed in the fur coat being bad conductor heat does not allow the body heat to flow outside. Hence we feel warmer in a fur coat. D YG (iii) Eskimos make double walled houses of the blocks of ice. Air enclosed in between the double walls prevents transmission of heat from the house to the cold surroundings. For exactly the same reason, two thin blankets are warmer than one blanket of their combined thickness. The layer of air enclosed in between the two blankets makes the difference. (iv) Wire gauze is placed over the flame of Bunsen burner U while heating the flask or a beaker so that the flame does not go beyond the gauze and hence there is no direct contact between the ST flame and the flask. The wire gauze being a good conductor of heat, absorb the heat of the flame and transmit it to the flask. Davy's safety lamp has been designed on this principle. The gases in the mines burn inside the gauze placed around the flame of the lamp. The temperature outside the gauze is not high, so the gases outside the gauze do not catch fire. (v) Birds often swell their feathers in winter. By doing so, they enclose more air between their bodies and the feathers. The air, being bad conductor of heat prevents the out flow of their body heat. Thus, birds feel warmer in winter by swelling their feathers. (6) Relation between temperature gradient and thermal conductivity 132 Transmission of Heat dQ d   KA dt dx In steady state, rate of flow of heat If = – KA (Temperature gradient) 1 dQ is constant then temperature gradient  dt K 60 Temperature difference between the hot end and the cold end in steady state is inversely proportional to K, i.e. in case of good conductors temperature of the cold end will be very near to hot end. E3 In ideal conductor where K = , temperature difference in steady state will be zero. (7) Wiedmann-Franz law At a given temperature T, the ratio of thermal conductivity to electrical conductivity is constant i.e., (K / T ) = constant, i.e., a substance which is a good conductor of heat (e.g., silver) ID is also a good conductor of electricity. Mica is an exception to above law. (8) Thermometric conductivity or diffusivity U It is a measure of rate of change of temperature (with time) when the body is not in steady state (i.e., in variable state) D YG The thermometric conductivity or diffusivity is defined as the ratio of the coefficient of thermal conductivity to the thermal capacity per unit volume of the material. mc = c V Thermal capacity per unit volume =  Diffusivity (D) = (As  is density of substance) K c Unit : m2/sec and Dimension : [L2 T 1 ] U (9) Thermal resistance The thermal resistance of a body is a measure of its opposition to the flow of heat through ST it. It is defined as the ratio of temperature difference to the heat current (= Rate of flow of heat) Now, temperature difference = ( 1   2 ) and heat current, H =  Thermal resistance, R  Unit : o 1   2 H  1   2 Q/t  1   2 l  KA( 1   2 ) / l KA C  sec / cal or K  sec / kcal and Dimension : [M 1 L2T 3 ] 14.3 Electrical Analogy For Thermal Conduction. Q t Transmission of Heat 133 It is an important fact to appreciate that there exists an exact similarity between thermal and electrical conductivities of a conductor. Thermal conduction Electric charge flows from higher potential to lower potential Heat flows from higher temperature to lower temperature The rate of flow of charge is called the electric current, The rate of flow of heat may be called as heat current dq dt H i.e. where R is the electrical resistance of the conductor where  = Resistivity and  conductivity l A  where R is conductor the thermal resistance l A = Electrical D YG V  V2  A dq I 1  (V1  V2 ) dt R l of The thermal resistance may be defined as R  U The electrical resistance is defined as R  Similarly, the heat current may be related with  2 the temperature difference as H  1 R ID The relation between the electric current and the potential difference is given by Ohm's law, that is V  V2 I 1 R dQ dt E3 I i.e. 60 Electrical conduction the l KA where K = Thermal conductivity of conductor dQ    2 KA H 1  (1   2 ) dt R l Sample problems based on Conduction Problem 1. The heat is flowing through a rod of length 50 cm and area of cross-section 5cm2. Its ends are respectively at 25 oC and 125 oC. The coefficient of thermal conductivity of the material of the rod is 0.092 kcal/m×s× oC. The temperature gradient in the rod is U (a) 2 oC / cm ST Solution : (a) Temperature gradient  Problem 2. (b) 2 oC / m (c) 20 oC / cm (d) 10 oC /m   2  1 125  25  2 o C / cm.   x x 50 Consider two rods of same length and different specific heats (s1 and s2), conductivities K1 and K2 and areas of cross-section (A1 and A2) and both giving temperature T1 and T2 at their ends. If the rate of heat loss due to conduction is equal, then (a) K1 A1  K2 A2 (b) K2 A1  K1 A2 (c) K1 A1 K 2 A2  s1 s2 (d) K 2 A1 K1 A2  s2 s1  dQ   dQ  Solution : (a) According to problem, rate of heat loss in both rods are equal i.e.     dt   1  dt  2  given] K 1 A1  1 K A   2 2 2 l1 l2  K 1 A1  K 2 A 2 [As  1   2 = (T1 – T2) and l1  l 2 134 Transmission of Heat Problem 3. Two rods (one semi-circular and other straight) of same material and of same crosssectional area are joined as shown in the figure. The points A and B are maintained at different temperature. The ratio of the heat transferred through a cross-section of a semicircular rod to the heat transferred through a cross-section of the straight rod in a given time is [UPSEAT 2002] (b) 1 : 2 60 (a) 2 :  Straight rod A (c)  : 2 dQ KA  , For both rods K, A and  are same   dt l So Problem 4. (dQ / dt ) semi circular (dQ / dt ) straight  l straight l semicircular  2r 2 . r  dQ 1  dt l ID Solution : (a) E3 (d) 3 : 2 B For cooking the food, which of the following type of utensil is most suitable U [MNR 1986; MP PET 1990; CPMT 1991; SCRA 1998;MP PMT/PET 1998, 2000; RPET 2001] (a) High specific heat and low conductivity (b) High specific heat and high conductivity D YG (c) Low specific heat and low conductivity (d) Low specific heat and high conductivity Solution : (d) Cooking utensil should conduct maximum and absorb minimum heat so it should possess high conductivity and low specific heat. Problem 5. A heat flux of 4000 J/s is to be passed through a copper rod of length 10 cm and area of cross-section 100 cm2. The thermal conductivity of copper is 400 W/moC. The two ends of this rod must be kept at a temperature difference of (a) 1 oC dQ KA   dt l U Solution : (c) From    l dQ  K  A dt (c) 100 oC  0.1 400  (100  10  4 ) (d) 1000 oC  4000 = 100oC The coefficients of thermal conductivity of copper, mercury and glass are respectively Kc, ST Problem 6. (b) 10 oC Km and Kg such that Kc > Km > Kg. If the same quantity of heat is to flow per second per unit area of each and corresponding temperature gradients are respectively Xc, Xm and Xg then (a) Xc= Xm = Xg Solution : (c) dQ / dt     K  A  x  (b) Xc> Xm > Xg  (c) Xc Kg therefore X C  X m  X g. Problem 7. A room is maintained at 20oC by a heater of resistance 20 ohm connected to 200 volt mains. (a) 15.24 oC (b) 15.00 oC 60 The temperature is uniform through out the room and heat is transmitted through a glass window of area 1 m2 and thickness 0.2 cm. What will be the temperature outside? Given that thermal conductivity K for glass is 0.2 cal/m  oC  sec and J = 4.2 J/cal (c) 24.15 oC (d) None of the above Solution : (a) As the temperature of room remain constant therefore the rate of heat generation from the heater should be equal to the rate of flow of heat through a glass window V2   R    t   KA.t  l  E3 1 J 1 (200 ) 2 0. 2  1  (20   )     = 15.24oC 4.2 20 0. 2  10  2 outside] Problem 8. [where  = temperature of ID  A point source of heat of power P is placed at the centre of a spherical shell of mean radius 2R 2 KT P (b) 4R 2 KT P Solution : (b) Rate of flow of heat or power (P)   Thickness of shell x  Problem 9. (c) R 2 KT (d) P D YG (a) U R. The material of the shell has thermal conductivity K. If the temperature difference between the outer and the inner surface of the shell is not to exceed T, then the thickness of the shell should not be less than R 2 KT 4P KA  K 4R 2 T  x x 4R 2 KT. P There are three thermometers – one in contact with the skin of the man, other in between U the vest and the shirt and third in between the shirt and coat. The readings of the thermometers are 30oC, 25oC and 22oC respectively. If the vest and shirt are of the same thickness, the ratio of their thermal conductivities is ST (a) 9 : 25 (b) 25 : 9 (c) 5 : 3 (d) 3 : 5 Solution : (d) Rate of flow of heat will be equal in both vest and shirt   shirt Kvest Kvest A. vest t K shirtA shirtt    K shirt vest l l  Kvest 25  22 3  . K shirt 30  25 5 14.4 Combination of Conductors. (1) Series combination : Let n slabs each of cross-sectional area A, lengths K1 , K 2 , K 3...... Kn respectively be connected in the series Heat current is the same in all the conductors. 1 l1 , l2 , l3 ,......ln 2 3 4 and conductivities n – 1 n K1 K2 K3 Kn l1 l2 l3 ln 136 Transmission of Heat i.e., Q  H1  H 2  H 3.........  H n t (i) Equivalent resistance R  R1  R 2  R 3 ..... R n (ii) If K s is equivalent conductivity, then from relation R  l KA Ks  l1  l 2  l 3 ...... ln l1 l l l  2  3 ........ n K1 K 2 K 3 Kn ID  E3 l1  l 2  l3 .....ln l l l l  1  2  3 ....  n Ks K1 A K 2 A K 3 A Kn A 60 K1 A( 1   2 ) K 2 A( 2   3 ) K 3 A( 3   4 ) K A( n 1   n )   ........  n l1 l2 l3 ln (iii) Equivalent thermal conductivity for n slabs of equal length K  2 K1 K 2 K1  K 2 U For two slabs of equal length, K  n 1 1 1 1   ..... K1 K 2 K 3 Kn D YG (iv) Temperature of interface of composite bar : Let the two bars are arranged in series as shown in the figure. Then heat current is same in the two conductors. i.e., Q K1 A(1   ) K 2 A(   2 )   t l1 l2  K1 l1 2 K2 l2 U K1 K 1  2  2 l l2 By solving we get   1 K1 K 2  l1 l2 1 ST If (l1  l2  l) then   K11  K 2 2 K1  K 2 (2) Parallel Combination Let n slabs each of length l, areas A1 , A2 , A3 ,..... An and thermal conductivities K1 , K 2 , K 3 ,..... Kn are connected in parallel then. (i) Equivalent resistance 1 1 1 1 1    ..... R R1 R 2 R 3 Rn (ii) Temperature gradient across each slab will be same. (iii) Heat current in each slab will be different. Net heat current will be the sum of heat currents through individual slabs. i.e., H  H 1  H 2  H 3 .... H n K( A1  A 2  A 3 .....  An )( 1   2 ) K1 A1 (1   2 ) K 2 A2 (1   2 ) K 3 A3 (1   2 ) K A (   2 )   .......  n n 1 l l l l l Transmission of Heat 137  K K 1 A1  K 2 A 2  K 3 A 3 ..... K n An A1  A 2  A 3 ..... An 1 A1 K1 K2 K  K 2  K 3 ..... K n For n slabs of equal area K  1 n A2 A3 K3 K1  K 2 2 60 Equivalent thermal conductivity for two slabs of equal area K  2 l A3 Kn Sample problems based on Combination of conductors E3 Problem 10. Two rods of same length and material transfer a given amount of heat in 12 seconds, when they are joined end to end. But when they are joined lengthwise, then they will transfer same heat in same conditions in [BHU 1998; UPSEAT 2002] .t l t  l A  l   2A t1 l A  1  2   1    1 t2 l 2 A1  l1 / 2   A1 (d) 48 s [As Q, K and  are constant]    D YG t1 t 12  4  t2  1  3 t2 4 4 (c) 1.5 s K U Solution : (b) Q  K A (b) 3 s ID (a) 24 s K K K Problem 11. Three rods made of the same material and having the same cross section have been joined as shown in the figure. Each rod is of the same length. The left and right ends are kept at 0oC and 90 o respectively. The temperature of the junction of the three rods will be (a) 45o C 90oC B 0o C (b) 60o C U A C (c) 30o C 90oC ST (d) 20o C Solution : (b) Let the conductivity of each rod is K. By considering the rods B and C are in parallel, effective thermal conductivity of B and C will be 2K. Now with the help of given formula K   K 2 2 Temperature of interface   1 1 K1  K 2    0o C K 90oC 2K K  0  2 K  90 180   60 o C. K  2K 3 Problem 12. Three rods of same dimensions are arranged as shown in figure they have thermal conductivities K1, K2 and K3. The points P and Q are maintained at different temperatures 138 Transmission of Heat for the heat to flow at the same rate along PRQ and PQ then which of the following option is correct [KCET (Engg. & Med.) 2001] (a) K 3  1 (K 1  K 2 ) 2 R (b) K3  K1  K2 K2 60 (c) K 3  K1 K1 K 2 K1  K 2 P Solution : (c) Rate of flow of heat along PQ E3 (d) K3  2 (K1  K2 ) Q K3 K A   dQ   3   l  dt  PQ.….(i) ID K A   dQ  Rate of flow of heat along PRQ   s  2l  dt  PRQ Effective conductivity for series combination of two rods of same length K s  2 K 1 K 2 A K 1 K 2 A  dQ  . .   K1  K 2 l  dt  PRQ K 1  K 2 2l K3 .….(ii) K1 K 2 K1  K 2 D YG Equating (i) and (ii) U So 2 K1 K 2 K1  K 2 Problem 13. The coefficient of thermal conductivity of copper is nine times that of steel. In the composite cylindrical bar shown in the figure. What will be the temperature at the junction of copper and steel [MP PMT 2000] (a) 75oC 100oC 0o C Copper U (b) 67oC (c) 33oC 18 cm Steel 6 cm ST (d) 25oC Solution : (a) K1 = 9K2, l1 = 18 cm, l2 = 6 cm, 1 = 100oC, 2 = 0oC K1 K 9K2 K 1  2  2 100  2  0 l l2 50  0 6 Temperature of the junction   1    18   75 o C 9K2 K2 K1 K 2 8 / 12   l1 l2 18 6 Problem 14. A cylinder of radius R made of a material of thermal conductivity K1 is surrounded by a cylindrical shell of inner radius R and outer radius 2R made of material of thermal conductivity K2. The two ends of the combined system are maintained at two different temperatures. There is no loss of heat across the cylindrical surface and the system is in steady state. The effective thermal conductivity of the system is Transmission of Heat 139 [IIT-JEE 1988; MP PMT 1994, 97; SCRA 1998] (a) K1  K2 (b) K1 K 2 K1  K 2 (c) K1  3 K 2 4 (d) 3 K1  K 2 4 Solution : (c) We can consider this arrangement as a parallel combination of two materials having different thermal conductivities K1 and K2 K 1 A1  K 2 A 2 A1  A 2 60 For parallel combination K  K2 A1 = Area of cross-section of internal cylinder = R2, R K1 2R  K 1.R 2  K 2 3R 2 K R  3R 2 2  E3 A2 = Area of cross-section of outer cylinder =  (2R)2 – (R)2 = 3R2 K1  3 K 2 4 Problem 15. The temperature of the interface of a compound wall as shown in the figure, in terms of (a) ID their thermal resistances R1 and R2 is 1   2 R1 1  R 2 2 R1  R 2 D YG (c) (d) 2 U 2 R1 2  R 2 1 (b) R1  R 2 1 K1 K2 2 > 1 R 2 1  R1 2 1   2 Solution : (b) Temperature of interface   Substituting K 1  l R1 A K 1 1  K 2 2 K1  K 2 and K 2  l R2 A we get   R1 2  R 2 1. R1  R 2 U Problem 16. Six identical conducting rods are joined as shown in figure. Points A and D are maintained ST at temperatures 200oC and 20oC respectively. The temperature of junction B will be (a) 120oC 20oC 200oC o (b) 100 C A C B D o (c) 140 C (d) 80oC Solution : (c) Let the thermal resistance of each rod is R R Effective thermal resistance between B and D = 2R Temperature of interface   R1 2  R 2 1 R1  R 2 R R A R B C R R D 140 Transmission of Heat   R  20  2 R  200 420   140 o C. R  2R 3 R 200oC A  B 2R 20oC D 14.5 Ingen-Hauz Experiment. It is used to compare thermal conductivities of different materials. If l1 and l2 are the K1 l12  K 2 l22 60 lengths of wax melted on rods, then the ratio of thermal conductivities is i.e., in this experiment, we observe Thermal conductivity  (length)2 E3 14.6 Searle's Experiment. It is a method of determination of K of a metallic rod. Here we are not much interested in ID the detailed description of the experimental setup. We will only understand its essence, which is the essence of solving many numerical problems. In this experiment a temperature difference ( 1   2 ) is maintained across a rod of length l and area of cross section A. If the thermal conductivity of the material of the rod is K, then the amount of heat transmitted by the rod from the hot end to the cold end in time t is given by, KA(1   2 ) t l U Q......(i) D YG In Searle's experiment, this heat reaching the other end is utilized to raise the temperature of certain amount of water flowing through pipes circulating around the other end of the rod. If temperature of the water at the inlet is  3 and at the outlet is  4 , then the amount of heat absorbed by water is given by, Q  mc ( 4   3 )......(ii) Where, m is the mass of the water which has absorbed this heat and temperature is raised and c is the specific heat of the water U Equating (i) and (ii), K can be determined i.e., K  numericals we may have the situation where the amount of heat travelling to ST Note :  In mc ( 4   3 ) l A (1   2 ) t the other end may be required to do some other work e.g., it may be required to melt the given amount of ice. In that case equation (i) will have to be equated to mL. i.e. mL  KA(1   2 ) t l 14.7 Growth of Ice on Lake. Water in a lake starts freezing if the atmospheric temperature drops below 0 o C. Let y be the thickness of ice layer in the lake at any instant t and atmospheric temperature is   o C. The Transmission of Heat 141 temperature of water in contact with lower surface of ice will be zero. If A is the area of lake, heat escaping through ice in time dt is dQ1  KA[0  ( )] dt y 60 Now, suppose the thickness of ice layer increases by dy in time dt, due to escaping of above heat. Then – dQ 2  mL   (dy A) L °C Ice y dy E3 As dQ 1  dQ 2 , hence, rate of growth of ice will be (dy / dt )  (K / Ly ) L So, the time taken by ice to grow to a thickness y is t  K L 0 y dy  2 K y y L K ID If the thickness is increased from y 1 to y 2 then time taken t   Air y2 y1 Wate r 2 ydy  0°C 4°C L 2 (y 2  y 12 ) 2 K (i) Take care and do not apply a negative sign for putting values of temperature in formula U and also do not convert it to absolute scale. (ii) Ice is a poor conductor of heat, therefore the rate of increase of thickness of ice on ponds decreases with time. D YG (iii) It follows from the above equation that time taken to double and triple the thickness, will be in the ratio of t1 : t 2 : t3 :: 12 : 2 2 : 3 2 , i.e., t1 : t 2 : t3 :: 1 : 4 : 9 (iv) The time intervals to change the thickness from 0 to y, from y to 2y and so on will be in the ratio t1 : t 2 : t3 :: (12  0 2 ) : (2 2  12 ) : (3 2 : 2 2 ) ; t1 : t 2 : t 3 :: 1 : 3 : 5 U Sample problems (miscellaneous) based on Conduction Problem 17. If the ratio of coefficient of thermal conductivity of silver and copper is 10 : 9, then the ST ratio of the lengths upto which wax will melt in Ingen Hausz experiment will be (a) 6 : 10 (b) 10 : 3 Solution : (b) According to Ingen Hausz, K  l2 (c) 100 : 81  l1  l2 K1  K2 10  9 (d) 81 : 100 10. 3 Problem 18. An ice box used for keeping eatables cool has a total wall area of 1 metre2 and a wall thickness of 5.0 cm. The thermal conductivity of the ice box is K= 0.01 J/moC. It is filled with ice at 0oC along with eatables on a day when the temperature is 30 oC. The latent heat of fusion of ice is 334×103 J/kg. The amount of ice melted in one day is (1 day = 86,400 seconds) [MP PMT 1995] (a) 776 g (b) 7760 g (c) 11520 g (d) 1552 g 142 Transmission of Heat Solution : (d) Quantity of heat transferred through wall will be utilized in melting of ice. Q   Amount of ice melted m  0.01  1  (30  0)  86400 5  10  2  334  10 3 KA  t x L  1.552 kg or 1552g 60 m KA  t  mL x Problem 19. Ice starts forming in lake with water at 0oC and when the atmospheric temperature is – 10oC. If the time taken for 1 cm of ice be 7 hr, then the time taken for the thickness of ice to change from 1 cm to 2 cm is (a) 7 hrs (b) 14 hrs E3 [NCERT 1971; MP PMT / PET 1988] (c) Less than 7 hrs Solution : (d) Time required in increment of thickness from y1 to y2 t  y1 = 0, y2 = 1 cm In second condition y1 = 1 cm, y2 = 2 cm t1 1  t 2 3  L 2 (y 2  y12 ) 2 K t1  (12 – 02) then t2  (22 – 12) ID In first condition (d) More than 7 hrs then   t2 = 3  t1 = 3  7 = 21 hrs. U Problem 20. The only possibility of heat flow in a thermos flask is through its cork which is 75 cm2 in area D YG and 5 cm thick. Its thermal conductivity is 0.0075 cal/cm secoC. The outside temperature is 40oC and latent heat of ice is 80 cal g–1. Time taken by 500 g of ice at 0oC in the flask to melt into water at 0oC is [CPMT 1974, 78; MNR 1983] (a) 2.47 hr (b) 4.27 hr (c) 7.42 hr (d) 4.72 hr. KA  t x  500  80  U Solution : (a) mL  0.0075  75  (40  0)t 5  t = 8.9  103 sec = 2.47 hr. Problem 21. There is ice formation in a tank of water of thickness 10 cm. How much time it will take to ST have a layer of 0.1 cm below it ? The outer temperature is – 5oC, the thermal conductivity of ice K = 0.005 cal/cm-secoC, the latent heat of ice is 80 cal/gm and the density of ice is 0.91 gm/cc (a) 46.39 minutes Solution : (c) t   (b) 47.63 minutes (c) 48.77 minutes  l 2 0. 91  80 y 2  y 12  [(10. 1) 2  (10 ) 2 ]  2926 sec  48.77 min. 2k  2  0. 005  5 14.8 Convection. Mode of transfer of heat by means of migration of material particles of medium is called convection. It is of two types. (d) 49.31 minutes Transmission of Heat 143 (1) Natural convection : This arise due to difference of densities at two places and is a consequence of gravity because on account of gravity the hot light particles rise up and cold heavy particles try setting down. It mostly occurs on heating a liquid/fluid. convection process is called forced convection. In this case Newton's law of cooling holds good. According to which rate of loss of heat from a hot body due to moving fluid is directly proportional to the surface area of body and excess temperature E3 of body over its surroundings Q  A(T  T0 ) t i.e. 60 (2) Forced convection : If a fluid is forced to move to take up heat from a hot body then the ID Q  h A(T  T0 ) t where h = Constant of proportionality called convection coefficient, T = Temperature of body and T 0 = Temperature of surrounding U Convection coefficient (h) depends on properties of fluid such as density, viscosity, specific heat and thermal conductivity. D YG (i) Natural convection takes place from bottom to top while forced convection in any direction. (ii) In case of natural convection, convection currents move warm air upwards and cool air downwards. That is why heating is done from base, while cooling from the top. (iii) Natural convection plays an important role in ventilation, in changing climate and weather and in forming land and sea breezes and trade winds. U (iv) Natural convection is not possible in a gravity free region such as a free falling lift or an orbiting satellite. ST (v) The force of blood in our body by heart helps in keeping the temperature of body constant. (vi) If liquids and gases are heated from the top (so that convection is not possible) they transfer heat (from top to bottom) by conduction. (vii) Mercury though a liquid is heated by conduction and not by convection. 14.9 Radiation. The process of the transfer of heat from one place to another place without heating the intervening medium is called radiation. Precisely it is electromagnetic energy transfer in the form of electromagnetic wave through any medium. It is possible even in vacuum. 144 Transmission of Heat For example, the heat from the sun reaches the earth through radiation. Properties of thermal radiation (1) The wavelength of thermal radiations ranges from 7.8  10 7 m to 4  10 4 m. They belong Frequency Cosmic rays > 1021 Hz Gamma rays 1018 - 1021 Hz X-rays 1016 - 1019 Hz 10–13 - 10–10 m 10–11 - 10–8 m rays (Heat) 4  1014 - 7.5  1014 Hz 4  10–7 - 7.8  10–7 m Å) 3  1011 - 4  1014 Hz 7.8  10–7 - 10–3 Å) 3  108 - 3  1011 Hz D YG Microwaves Radio waves 1.4  10–8 - 4  10–7 m ID 7.5  1014 - 2  106 Hz Visible rays  Infrared < 10–13 m U Ultraviolet rays Wavelength E3 Radiation 60 to infra-red region of the electromagnetic spectrum. That is why thermal radiations are also called infra-red radiations. 104 - 3  109 Hz (0.1 Å - 100 Å) (140 Å - 4000 Å) (4000 Å - 7800 (7800 Å - 3  105  10–3 m - 0.1 m 0.1 m - 104 m (2) Medium is not required for the propagation of these radiations. (3) They produce sensation of warmth in us but we can’t see them. (4) Every body whose temperature is above zero Kelvin emits thermal radiation. U (5) Their speed is equal to that of light i.e. ( 3  10 8 m / s). (6) Their intensity is inversely proportional to the square of distance of point of observation from ST the source (i.e. I  1 / d 2 ). (7) Just as light waves, they follow laws of reflection, refraction, interference, diffraction and polarisation. (8) When these radiations fall on a surface then exert pressure on that surface which is known as radiation pressure. (9) While travelling these radiations travel just like photons of other electromagnetic waves. They manifest themselves as heat only when they are absorbed by a substance. (10) Spectrum of these radiations can not be obtained with the help of glass prism because it absorbs heat radiations. It is obtained by quartz or rock salt prism because these materials do Transmission of Heat 145 not have free electrons and interatomic vibrational frequency is greater than the radiation frequency, hence they do not absorb heat radiations. 14.10 Some Definition About Radiations. (1) Diathermanous Medium : A medium which allows heat radiations to pass through it 60 without absorbing them is called diathermanous medium. Thus the temperature of a diathermanous medium does not increase irrespective of the amount of the thermal radiations passing through it e.g., dry air, SO 2 , rock salt (NaCl). E3 (i) Dry air does not get heated in summers by absorbing heat radiations from sun. It gets heated through convection by receiving heat from the surface of earth. (ii) In winters heat from sun is directly absorbed by human flesh while the surrounding air ID being diathermanous is still cool. This is the reason that sun’s warmth in winter season appears very satisfying to us. (2) Athermanous medium : A medium which partly absorbs heat rays is called a thermous U medium As a result temperature of an athermanous medium increases when heat radiations pass through it e.g., wood, metal, moist air, simple glass, human flesh etc. D YG 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 100°C. 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. (3) Reflectance, Absorptance and transmittance U When thermal radiations (Q) fall on a body, they are partly reflected, partly absorbed and partly transmitted. (i) Reflectance or reflecting power (r) : It is defined as the ratio of the amount of thermal ST radiations reflected (Qr) by the body in a given time to the total amount of thermal radiations incident on the body in that time. (ii) Absorptance or absorbing power (a) : It is defined as the ratio of the amount of thermal radiations absorbed (Qa) by the body in a given time to the total amount of thermal radiatons incident on the body in that time. (iii) Transmittance or transmitting power (t) : It is defined as the ratio of the amount of thermal radiations transmitted (Qt) by the body in a given time to the total amount of thermal radiations incident on the body in that time. From the above definitions r Qr Q , a a Q Q and t  Qt Q 146 Transmission of Heat rat  By adding we get  Q r Q a Q t (Q r  Q a  Q t )    1 Q Q Q Q rat 1 (b) For perfect reflector : r = 1, a = 0 and t = 0 (c) For perfect absorber : a = 1, r = 0 and t = 0 60 (a) r, a and t all are the pure ratios so they have no unit and dimension. (Perfectly black body) (d) For perfect transmitter : t = 1, a = 0 and r = 0  r  a  1 or a  1  r E3 (e) If body does not transmit any heat radiation, t = 0 So if r is more, a is less and vice-versa. It means good reflectors are bad absorbers. (4) Monochromatic Emittance or Spectral emissive power ID For a given surface it is defined as the radiant energy emitted per sec per unit area of the 1 1   surface with in a unit wavelength around  i.e. lying between     to    . 2 2   Joule and Dimension : [ML1 T 3 ] D YG Unit : m  sec  Å 2 Energy Area  time  wavelength U Spectral emissive power (E  )  (5) Total emittance or total emissive power It is defined as the total amount of thermal energy emitted per unit time, per unit area of the body for all possible wavelengths. Watt Joule or 2 m  sec m2 U Unit : and E   0 E d Dimension : [MT 3 ] ST (6) Monochromatic absorptance or spectral absorptive power It is defined as the ratio of the amount of the energy absorbed in a certain time to the total heat energy incident upon it in the same time, both in the unit wavelength interval. It is dimensionless and unit less quantity. It is represented by a. (7) Total absorptance or total absorpting power : It is defined as the total amount of thermal energy absorbed per unit time, per unit area of the body for all possible wavelengths. a   0 a d It is also unit less and dimensionless quantity. Transmission of Heat 147 (8) Emissivity (e) : Emissivity of a body at a given temperature is defined as the ratio of the total emissive power of the body (Epractical ) to the total emissive power of a perfect black body (Eblack) at that temperature. E practical Eblack 60 e i.e. e = 1 for perfectly black body but for practical bodies emissivity (e) lies between zero and one (0 < e < 1). E3 (9) Perfectly black body : A perfectly black body is that which absorbs completely the radiations of all wavelengths incident on it. As a perfectly black body neither reflects nor transmits any radiation, therefore the absorptance of a perfectly black body is unity i.e. t = 0 and r = 0  a = 1. ID We know that the colour of an opaque body is the colour (wavelength) of radiation reflected by it. As a black body reflects no wavelength so, it appears black, whatever be the colour of radiations incident on it. U When perfectly black body is heated to a suitable high temperature, it emits radiation of all D YG possible wavelengths. For example, temperature of the sun is very high (6000 K approx.) it emits all possible radiation so it is an example of black body. (10) Ferry’s black body : A perfectly black body can’t be realised in practice. The nearest example of an ideal black body is the Ferry’s black body. It is a O ST U doubled walled evacuated spherical cavity whose inner wall is blackened. There is a fine hole in it. All the radiations incident upon this hole are absorbed by this black body. If this black body is heated to high temperature then it emits radiations of all wavelengths. Sample problems based on Radiation Problem 22. An ideal black body at room temperature is thrown into a furnace. It is observed that (a) Initially it is the darkest body and at later times the brightest (b) It is the darkest body at all times (c) It cannot be distinguished at all times (d) Initially it is the darkest body and at later times is cannot be distinguished Solution : (d) Problem 23. A body is in thermal equilibrium with the surrounding (a) It will stop emitting heat radiation (b) Amount of radiations emitted and absorbed by it will be equal P 148 Transmission of Heat (c) It will emit heat radiations at faster rate (d) It will emit heat radiations slowly Solution : (b) Problem 24. If transmission power of a surface is 1/6 and reflective power is 1/3, then its absorptive Solution : (b) t  1 3 1 , 6 (b) r 1 3 1 2 (c) 1 6 and we know a  r  t  1  a = 1– r– t 1 (d) 1 12 E3 (a) 60 power will be 1 1 1  . 3 6 2 14.11 Prevots Theory of Heat Exchange. ID (1) Every body emits heat radiations at all finite temperature (Except 0 K) as well as it absorbs radiations from the surroundings. (2) Exchange of energy along various bodies takes place via radiation. U (3) The process of heat exchange among various bodies is a continuous phenomenon. D YG (4) If the amount of radiation absorbed by a body is greater than that emitted by it then the temperature of body increases and it appears hotter. (5) If the amount of radiation absorbed by a body is less than that emitted by it, then the temperature of the body decreases and consequently the body appears colder. (6) If the amount of radiation absorbed by a body is equal to that emitted by the body, then the body will be in thermal equilibrium and the temperature of the body remains constant. U (7) At absolute zero temperature (0 K or – 273°C) this law is not applicable because at this temperature the heat exchange among various bodies ceases. 14.12 Kirchoff's Law. ST The ratio of emissive power to absorptive power is same for all surfaces at the same temperature and is equal to the emissive power of a perfectly black body at that temperature. Thus if apractical and Epractical represent the absorptive and emissive power of a given surface, while ablack and Eblack for a perfectly black body, then according to law But for a perfectly black body ablack = 1 so E practical a practical E practical a practical  E black a black  E black If emissive and absorptive powers are considered for a particular wavelength ,  E   a    (E  ) black  practical Transmission of Heat 149 Now since (E)black is constant at a given temperature, according to this law if a surface is a good absorber of a particular wavelength it is also a good emitter of that wavelength. This in turn implies that a good absorber is a good emitter (or radiator) Applications of Kirchoff's law 60 (1) Sand is rough black, so it is a good absorber and hence in deserts, days (when radiation E3 from the sun is incident on sand) will be very hot. Now in accordance with Kirchoff’s law, good absorber is a good emitter so nights (when sand emits radiation) will be cold. This is why days are hot and nights are cold in desert. (2) Sodium vapours, on heating, emit two bright yellow lines. These are called D1, D2 lines of sodium. When continuos white light from an arc lamp is made to pass through sodium vapours at low temperature, the continuous spectrum is intercepted by two dark lines exactly in ID the same places as D1 and D2 lines. Hence sodium vapours when cold, absorbs the same wavelength, as they emit while hot. This is in accordance with Kirchoff's law. (3) When a shining metal ball having some black spots on its surface is heated to a high temperature and is seen in dark, the black spots shine brightly and the shining ball becomes U dull or invisible. The reason is that the black spots on heating absorb radiation and so emit these in dark while the polished shining part reflects radiations and absorb nothing and so does D YG not emit radiations and becomes invisible in the dark. (4) When a green glass is heated in furnace and taken out, it is found to glow with red light. This is because red and green are complimentary colours. At ordinary temperatures, a green glass appears green, because it transmits green colour and absorb red colour strongly. According to Kirchoff's law, this green glass, on heating must emit the red colour, which is absorbed strongly. Similarly when a red glass is heated to a high temperature it will glow with green light. U (5) Kirchoff' law also explains the existence of Fraunhoffer lines. These are some dark lines ST observed in the otherwise spectrum of the sun. According to Fraunhoffer, the central portion of the sun, called photosphere, is at a very high temperature and emits continuous light of all wavelengths. Before reaching us, the light passes through outer portion of the sun, called chromosphere. The chromosphere has some terrestrial elements in vapour form at lower Photo sphere Chromospher e Radiations of wavelengths ranging 0 to  Radiations having all wavelengths except wavelengths absorbed by various elements in chromosphere temperature than that of photosphere. These elements absorb those wavelength which they would emit while hot. These absorbed wavelengths, which are missing appear as dark lines in the spectrum of the sun. But during total solar eclipse these lines appear bright because the gases and vapour present in the chromosphere start emitting those radiation which they had absorbed. 150 Transmission of Heat (6) A person with black skin experiences more heat and more cold as compared to a person of white skin because when the outside temperature is greater, the person with black skin 60 absorbs more heat and when the outside temperature is less the person with black skin radiates more energy. Sample problems based on Kirrchoff's law Problem 25. The graph. Shown in the adjacent diagram, represents the variation of temperature (T) of E3 two bodies, x and y having same surface area, with time (t) due to the emission of radiation. Find the correct relation between the emissivity (e) and absorptivity (a) of the two bodies [IIT-JEE (Screening) 2003] (a) e x  e y & a x  ay T x t U (c) e x  e y & a x  ay (d) e x  e y & a x  ay y ID (b) e x  e y & a x  ay D YG Solution : (c) From the graph it is clear that initially both the bodies are at same temperature but after that at any instant temperature of body x is less then the temperature of body y. It means body x emits more heat i.e. emissivity of body x is more than body y  e x  e y and according to Kirchoff's law good emitter are also good absorber so a x  a y. Problem 26. Certain substance emits only the wavelengths 1 , 2 , 3 and 4 when it is at a high temperature. When this substance is at a colder temperature, it will absorb only the following wavelengths [MP PET 1990] U (a) 1 (b) 2 (d) 1 , 2 , 3 and 4 (c) 1 and 2 ST Solution : (d) If a body emits wavelength 1, 2, 3 and 4 at a high temperature then at a lower temperature it will absorbs the radiation of same wavelength. This is in accordance with Kirchoff's law. Problem 27. The following figure shows two air-filled bulbs connected by a U-tube partly filled with alcohol. What happens to the levels of alcohol in the limbs X and Y when an electric bulb placed midway between the bulbs is lighted Black Painted (a) The level of alcohol in limb X falls while that in limb Y rises (b) The level of alcohol in limb X rises while that in limb Y falls X Bulb (c) The level of alcohol falls in both limbs (d) There is no change in the levels of alcohol in the two limbs Alcohol Y Transmission of Heat 151 Solution : (a) Black bulb absorbs more heat in comparison with painted bulb. So air in black bulb expands more. Hence the level of alcohol in limb X falls while that in limb Y rises. 14.13 Distribution of Energy in The Spectrum of Black Body. Langley and later on Lummer and Pringsheim investigated the distribution of energy 60 amongst the different wavelengths in the thermal spectrum of a black body radiation. The results obtained are shown in figure. From these curves it is clear that E3 (1) At a given temperature energy is not uniformly distributed among different wavelengths. (2) At a given temperature intensity of heat radiation T3 increases with wavelength, reaches a maximum at a particular wavelength and with further increase in wavelength it decreases. With increase in temperature wavelength m ID (3) E corresponding to most intense radiation decreases in such a way that m  T  constant. [Wien’s law] T2 (4) For all wavelengths an increase in temperature causes an T1 m 3 m 2 m 1 U increase in intensity. T3 > T 2 > T1  D YG (5) The area under the curve  E d will represent the total  intensity of radiation at a particular temperature. This area increases with rise in temperature of the body. It is found to be directly proportional to the fourth power of absolute temperature of the body, i.e.,  E  E d  T 4 [Stefan’s law] (6) The energy (Emax) emitted corresponding to the wavelength of maximum emission (m) U increases with fifth power of the absolute temperature of the black body i.e., E max  T 5 Sample problems based on Energy distribution graph ST Problem 28. The plots of intensity versus wavelength for three black bodies at temperatures T1, T2 and T3 respectively are as shown. Their temperature are such that T3 (a) T1 >T2 > T3 (b) T1 >T3 > T2 I T1 T2 (c) T2 >T3 > T1 (d) T3 >T2 > T1 Solution : (b) According to Wien's law  m  T2.  1 and from the figure (m )1  (m )3  (m ) 2 therefore T1 > T3 > T 152 Transmission of Heat Problem 29. The adjoining diagram shows the spectral energy density distribution E of a black body at two different temperatures. If the areas under the curves are in the ratio 16 : 1, the value of temperature T is [DCE 1999] TK E 2000 K 60 (a) 32,000 K (b) 16,000 K  (c) 8,000 K Solution : (d) Area under curve represents the emissive power of the body ….(i) 4 16  T  From (i) and (ii)    2000 1    ET  T    E 2000  2000  4 ….(ii) ID But from Stefan's law E  T4  ET AT 16 (given)   E 2000 A 2000 1 E3 (d) 4,000 K T  2  T = 4000K. 2000 U Problem 30. Following graph shows the correct variation in intensity of heat radiations by black body and frequency at a fixed temperature D YG UV Visible Infra-red (a) E (b) UV Visible Infra-red E 3500K 1500K 2500K 2500K 1500K 3500K   Infra-red Visible Ultra-violet (c) ST U E Infra-red Visible Ultra-violet (d) E 3500K 1500K 2500K 2500K 1500K 3500K   Solution : (c) As the temperature of body increases, frequency corresponding to maximum energy in radiation (vm) increases this is shown in graph (c). 14.14 Wien's Displacement Law. When a body is heated it emits radiations of all wavelength. However the intensity of radiations of different wavelength is different. corresponding temperature to of maximum body (in intensity of radiation and Kelvin) is constant, i.e. m logm According to Wien's law the product of wavelength T log T Transmission of Heat 153 m T  b  constant Where b is Wien's constant and has value 2.89  10 3 m - K. 60 This law is of great importance in ‘Astrophysics’ as through the analysis of radiations coming from a distant star, by finding m the temperature of the star T ( b / m ) is determined. Sample problems based on Wien's displacement law Problem 31. A black body at 200 K is found to emit maximum energy at a wavelength of 14  m. When [MP PET 1991; BVP 2003] (a) 14  m (b) 70  F Solution : (c)  m T  constant  E3 its temperature is raised to 1000 K, the wavelength at which maximum energy is emitted is (c) 2.8  m (d) 2.8 mm  (m ) 2  (m )1 14  m = 2.8 m.  5 5 (m ) 2 T 200 1  1   (m )1 T2 1000 5 ID Problem 32. The energy spectrum of a black body exhibits a maximum around a wavelength 0. The temperature of the black body is now changed such that the energy is maximum around a 3 0. The power radiated by the black body will now increase by a factor of 4 (a) 256/81 U wavelength (b) 64/27 (c) 16/9 (d) 4/3 D YG Solution : (a) According to Wien's law wavelength corresponding to maximum energy decreases. When the temperature of black body increases i.e.  m T  constant  Now according to Stefan's law 4 T2  0 4  1   T1  2 30 / 4 3 E 2  T2  256 4.      E1  T1  3 81   4 Problem 33. The wavelength of maximum energy released during an atomic explosion was 2. 93  10 10 m. U Given that Wien's constant is 2.93  10 3 m –K, the maximum temperature attained must be of the order of (b) 107 K ST (a) 10–7 K [Haryana CEE 1996; MH CET 2002] Solution : (b) From Wien's displacement law m T  b (d) 5.86  10 7 K (c) 10–13 K  T  b m  2.93  10 3 2.93  10 10  10 7 K. Problem 34. Consider the following statements Assertion (A) : Blue star is at higher temperature than red star Reason (R) : Wien's displacement law states that T 1 m Of these statements (a) Both A and R are true and the R is a correct explanation of the A [AIIMS 2002] 154 Transmission of Heat (b) Both A and R are true but the R is not a correct explanation of the A (c) A is true but the R is false (d) Both A and R are false (e) A is false but the R is true 60 Solution : (a) Wavelength of radiation from blue star is less than that of red star. So its temperature will be higher than red star. Problem 35. A black body is at a temperature of 2880 K. The energy of radiation emitted by this object with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 (c) U1 > U2 Solution : (d) According to Wien's displacement law m T  b (d) U2 > U1 E ID m  b 2.88  10 6 nm-K    1000 nm T 2880 nm U2 U3 U1  (nm) U 499 500 i.e. energy corresponding to wavelength 1000 nm will be maximum i.e. U2 will be maximum U1 < U2 > U3 1499 1500 (b) U3 = 0 900 1000 (a) U1 = 0 E3 and between 1499 nm and 1500 nm is U3. The Wien's constant b  2.88  10 6 nm K. Then Energy distribution graph with wavelength will be as follows D YG Problem 36. Which of the following is the vm-T graph for a perfectly black body (a) A m [RPMT 1996] D B C (b) B (c) C (d) D Solution : (b) Wien's law  m  1 T A T or  m  T U m increases with temperature. So the graph will be straight line. 14.15 Law of Distribution of Energy. ST The theoretical explanation of black body radiation was done by Planck. If the walls of hollow enclosure are maintained at a constant temperature, then the inside of enclosure are filled with the electromagnetic radiation. The radiation coming out from a small hole in the enclosure are called black body radiation. According to Max Planck, the radiation inside the enclosure may be assumed to be produced by a number of harmonic oscillators. A harmonic oscillator oscillating with frequency  can possesses energies, which are integral multiples of h. Where h is a constant, called Planck's constant. Thus the harmonic oscillator can posses energies given by E  nh where n is an integer. Transmission of Heat 155 According to Planck's law E d  8hc  5 1 [e hc / KT  1] d hc   For radiations of short wavelength     KT   Planck's law reduces to Wien's energy distribution law E  d   A  5 e  B / T d  E3 hc   For radiations of long wavelength     KT   60 This law is valid for radiations of all wavelengths ranging from zero to infinite. Planck's law reduces to Rayleigh-Jeans energy distribution law E  d   4 d ID 14.16 Stefan's Law. 8KT E  T 4 or E  T 4 i.e. where  U According to it the radiant energy emitted by a perfectly black body per unit area per sec (i.e. emissive power of black body) is directly proportional to the fourth power of its absolute temperature, is a constant called Stefan’s constant having dimension [MT 3 4 ] and value D YG 5.67  10 8 W / m 2 K 4. (i) If e is the emissivity of the body then E  e T 4 (ii) If Q is the total energy radiated by the body then E  Q  e T 4  Q  A t e T 4 At (iii) If a body at temperature T is surrounded by a body at temperature T0, then Stefan's law may be put as E  e  (T 4  T04 ) U (iv) Cooling by radiation : If a body at temperature T is in an environment of temperature T0(< T), the body is loosing as well as receiving so net rate of loss of energy dQ  eA  (T 4  T04 ) dt ST …..(i) Now if m is the mass of body and c its specific heat, the rate of loss of heat at temperature T must be dQ dT  mc dt dt From equation (i) and (ii) …..(ii) mc dT  eA  (T 4  T04 ) dt  Rate of fall of temperature or rate of cooling, dT eA  4  (T  T04 ) dt mc i.e. when a body cools by radiation the rate of cooling depends on …..(iii) 156 Transmission of Heat (a) Nature of radiating surface i.e. greater the emissivity, faster will be the cooling. (b) Area of radiating surface, i.e. greater the area of radiating surface, faster will be the cooling. (c) Mass of radiating body i.e. greater the mass of radiating body slower will be the cooling. 60 (d) Specific heat of radiating body i.e. greater the specific heat of radiating body slower will be cooling. E3 (e) Temperature of radiating body i.e. greater the temperature of body faster will be cooling. (f) Temperature of surrounding i.e. greater the temperature of surrounding slower will be cooling. Sample problems based on Stefan's law ID Problem 37. Two black metallic spheres of radius 4m, at 2000 K and 1m at 4000 K will have ratio of energy radiation as (b) 4 : 1  Q1 A T  1  1 Q2 A 2  T2 4   r2   1  r22   T1   T2 4 (d) 2 : 1  1 4  2000        1:1.   16  1 4000 16      D YG Solution : (a) Q =  A t T4 (c) 1 : 4 U (a) 1 : 1 [RPET 2000; AIEEE 2002] 2 4 Problem 38. Two identical metal balls at temperature 200oC and 400oC kept in air at 27oC. The ratio of net heat loss by these bodies is (a) 1/ 4 (b) 1/ 2 (c) 1/ 16 (d) 473 4  300 4 673 4  300 4 Solution : (d) Emissive power of a body (T) in a surrounding (T0), E   (T 4  T04 ) or Q  (T14  T04 ) Q1 (T 4  T04 ) (473 ) 4  (300 ) 4  14 . Q 2 (T2  T04 ) (673 ) 4  (300 ) 4 U  Problem 39. Two spheres made of same material have radii in the ratio 1 : 2. Both are at same ST temperature. Ratio of heat radiation energy emitted per second by them is (a) 1 : 2 Q  A T 4 Solution : (c) P  t (b) 1 : 8 (c) 1 : 4 (d) 1 : 16 2 P A r2  1  1  1  1  12     P2 A2 2 4 r2   [If T = constant] Problem 40. Two spherical black bodies of radii r1 and r2 and with surface temperature T1 and T2 respectively radiate the same power. Then the ratio of r1 and r2 will be  T2  T1 (a)      2  T2  T1 (b)      4  T1  T2 (c)      2  T1  T2 (d)      4 Transmission of Heat 157 Solution : (a) P = A  T4 = 4r2 T4  P  r 2 T 4 or r2  1 T [As P = constant]  4 r1  T2  r2  T1 Problem 41. The rectangular surface of area 8 cm × 4 cm of a black body at a temperature of 127oC emits 3 E 8 (b) 81 E 16 (c) Solution : (d) Energy radiated by body per second T E2 l b   2  2   2 E1 l1 b1  T1 9 E 16 Q  A T 4 or t (d) Q  lbT4 t 81 E 64 [Area = l  b] E3 (a) 60 energy at the rate of E per second. If the length and breadth of the surface are each reduced to half of the initial value and the temperature is raised to 327oC, the rate of emission of energy will become [MP PET 2000] 4  (l / 2) (b1 / 2)  600  1 1 3   1        l1 b1 2 2 2  400   4 4  E2  81 E 64 ID Problem 42. A solid copper cube of edges 1 cm is suspended in an evacuated enclosure. Its temperature (a) 25 s (b) 50 s   (c) 200 s (d) 400 s e (6 a 2 ) dT eA  4 (T 4  T04 )  For the same fall in temperature, time dt  a  T  T04  3 dt mc (a   )c D YG Solution : (c) U is found to fall from 100oC to 99oC in 100s. Another solid copper cube of edges 2 cm, with similar surface nature, is suspended in a similar manner. The time required for this cube to cool from 100oC to 99oC will be approximately dt 2 a 2cm  2  dt 1 a1 1cm  dt2 = 2  dt1 = 2  100 sec = 200 sec [As A = 6a2 and m = V   = a3  ] Problem 43. Two metallic spheres S1 and S2 are made of the same material and have identical surface U finish. The mass of S1 is three times that of S2. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of S1 to that of S2 is (a) 1/ 3 dT e A  4  (T  T04 ) dt mc ST Solution : (b) (b) (1/ 3)1/ 3 (c) 1 / 3  Rate of cooling R  R1  1    R2  3  4 3 r   3  3 /1 R r 1  1  2 r R2 r1 r  1  4 3 4  r1    3  r23     r13  3r23   2     3 3   r1   3  But according to problem m1 = 3m2   Ratio of rate of cooling 4 r 2 (d) 1/3 1/3. Problem 44. A sphere, a cube and a thin circular plate, all made of same substance and all have same mass. These are heated to 200oC and then placed in a room, then the (a) Temperature of sphere drops to room temperature at last (b) Temperature of cube drops to room temperature at last 158 Transmission of Heat (c) Temperature of thin circular plate drops to room temperature at last (d) Temperature of all the three drops to room temperature at the same time Solution : (a) eA  4 dT eA  (T  T04 )  (T 4  T04 )  Vc dt mc  Rate of cooling R  A 60 [As masses are equal then volume of each body must be equal because materiel is same] i.e. rate of cooling depends on the area of cross-section and we know that for a given volume the area of cross-section will be minimum for sphere. It means the rate of cooling will be minimum in case of sphere. E3 So the temperature of sphere drops to room temperature at last. Problem 45. A solid copper sphere (density  and specific heat capacity c ) of radius r at an initial temperature 200K is suspended inside a chamber whose walls are at almost 0K. The time required (in  s) for the temperature of the sphere to drop to 100 K is 7 rc 72  (b) 27 rc 7  (c) (d) 7 rc 27  A dT  (T 4  T04 ) [In the given problem fall in temperature of body dT  (200  100 )  100 K dt mcJ U Solution : (b) 72 rc 7  ID (a) Temperature of surrounding T0 = 0K, Initial temperature of body T = 200K] r c J r c 4.2 7 r c 7 r c  10 6 s .  10 6   s ~–  s [As 48   48 80  72  D YG 100  4r 2  (200 4 3 dt r  c J 3 4  0 4 )  dt  J = 4.2] Problem 46. A sphere and a cube of same material and same volume are heated upto same temperature and allowed to cool in the same surroundings. The ratio of the amounts of radiations emitted will be 1/3   (c)   4 :1 3 (b) U (a) 1 : 1 6 (d) :1 1  4    2 3  2/3 :1 ST Solution : (c) Q =  A t (T4 – T04) If T, T0,  and t are same for both bodies then Q sphere Q cube  A sphere A cube  4r 2 6a 2 But according to problem, volume of sphere = Volume of cube  Substituting the value of a in equation (i) we get Q sphere Q cube  4r 2 6a 2  4r 2  4  1 / 3  6    r   3   14.17 Newton's Law of Cooling. 2  4r 2 4  6  3  2/3 r 2     6 1/3 :1 …..(i) 4 3 4  r  a 3  a     3 3  1/3 r Transmission of Heat 159 If in case of cooling by radiation the temperature T of body is not very different from that of surrounding T  T0  T T  4 T04  [(T0  T )  4 T04 ]  T04 4     T  4 T  1    1  T04 1   1  T0  T0     theorem]  4 T03 T …..(i) dT eA  4 By Stefan’s law,  [T  T04 ] dt mc or E3 0 dT eA   4 T03 T dt mc dT  T dt So  Cooling curve of a body 0 d   0 dt Time t ID From equation (i), [Using Binomial 60 i.e. U i.e., if the temperature of body is not very different from surrounding, rate of cooling is proportional to temperature difference between the body and its surrounding. This law is called Newton’s law of cooling. (1) Practical examples D YG (i) Hot water loses heat in smaller duration as compared to moderate warm water. (ii) Adding milk in hot tea reduces the rate of cooling. (2) Greater the temperature difference between body and its surrounding greater will be the rate of cooling. d  0 i.e. a body can never be cooled to a temperature lesser than its dt surrounding by radiation. U (3) If    0 , (4) If a body cools by radiation from  1o C to  2o C in time t, then and 1   2 ST    av  d  1   2  dt t 2    2     2   K 1 0  The Newton’s law of cooling becomes  1   t   2  This form of law helps in solving numericals. (5) Cooling curves : Temperatu re Curve between temperature of body  and time. Curve between rate of cooling (R) and temperature difference between body () and surrounding (0) R Tim e O ( – 0) 160 Transmission of Heat    0  Ae kt , which indicates temperature R  (   0 ). This is a straight line passing Curve between the rate of cooling (R) and body temperature (). Curve between log( – 0) and time K0  E3 R 60 through origin. loge ( – 0) decreases exponentially with increasing time. O R  K(   0 )  K  K 0 d d  (   0 )    Kdt dt (   0 ) ID This is a straight line intercept R-axis at  K 0 As t Integrating log e (   0 )   Kt  C U log e (   0 )   Kt  log e A This is a straight line with negative slope i.e. D YG (6) Determination of specific heat of a liquid : If volume, radiating surface area, nature of surface, initial temperature and surrounding of water and given liquid are equal and they are allowed to cool down (by radiation) then rate of loss of heat and fall in temperature of both will be same.  dQ   dQ       dt  water  dt  liquid  ms  W   m 1 s1  W     t2  t1    ST or ( 1   2 ) (   2 )  (m 1 s1  W ) 1 t1 t2 U (ms  W ) [where W = water equivalent of calorimeter] If density of water and liquid is  and  respectively then m  V and m'  V  Sample problems based on Newton's law of cooling Problem 47. A bucket full of hot water cools from 75 oC to 70oC in time T1, from 70oC to 65oC in time T2 and from 65oC to 60oC in time T3, then (a) T1  T2  T3 (b) T1  T2  T3 (c) T1  T2  T3 (d) T1  T2  T3 Solution : (c) According to Newton's law of cooling rate of cooling depends upon the difference of temperature between the body and the surrounding. It means that when the difference of temperature between the body and the surrounding is small then time required for same Transmission of Heat 161 fall in temperature is more in comparison with the same fall at higher temperature difference between the body and surrounding. So according to problem T1 < T2 < T3. Problem 48. A cup of tea cools from 80oC to 60oC in one minute. The ambient temperature is 30oC. In cooling from 60oC to 50oC it will take 1   2 Solution : (d) According to Newton's law of cooling For first condition (c) 90 Seconds t 80  60  80  60    30  60  2  and for second condition    2   1  2   …..(i) (d) 50 Seconds 60 (b) 60 Seconds E3 (a) 30 Seconds 60  50  60  50    30  t  2  ….(ii) ID By solving (i) and (ii) we get t = 48 sec ~– 50 sec. Problem 49. A body takes T minutes to cool from 620C to 610C when the surrounding temperature is 300C. The time taken by the body to cool from 460C to 45.50C is Equal to T minutes (c) Less than T minutes(d) U (a) Greater than T minutes (b) 1   2 D YG Solution : (b) According to Newton's law of cooling For first condition t    2   1  2   62  61  62  61    30  T  2  and for second condition …..(i) 46  45. 5  46  45. 5    30  t 2   ….(ii) By solving (i) and (ii) we get t = T sec. Problem 50. The rates of cooling of two different liquids put in exactly similar calorimeters and kept in U identical surroundings are the same if (a) The masses of the liquids are equal ST (b) Equal masses of the liquids at the same temperature are taken (c) Different volumes of the liquids at the same temperature are taken (d) Equal volumes of the liquids at the same temperature are taken Solution : (d) dT A 4  (T  T04 ). If the liquids put in exactly similar calorimeters and identical dt mc surrounding then we can consider T0 and A constant then (T 4  T04 ) dT  dt mc If we consider that equal masses of liquid (m) are taken at the same temperature then dT 1  dt c 162 Transmission of Heat So for same rate of cooling c should be equal which is not possible because liquids are of different nature. Again from (i) equation (T 4  T04 ) (T 4  T04 ) dT dT    dt V c dt mc 60 Now if we consider that equal volume of liquid (V) are taken at the same temperature then dT 1.  c dt E3 So for same rate of cooling multiplication of  c for two liquid of different nature can be possible. So option (d) may be correct. Problem 51. Hot water cools from 60oC to 50oC in the first 10 minutes and to 42oC in the next 10 minutes. The temperature of the surrounding (a) 5oC t    2   1  2   For first condition 60  50  60  50    10 2    1 = K [55 – ] ……(i) 50  42  50  42    10 2    0.8 = K (46 –  ) …..(ii) D YG For second condition (d) 20oC ID 1   2 (c) 15oC U Solution : (b) (b) 10oC From (i) and (ii) we get  = 10oC 14.18 Temperature of The Sun and Solar Constant. If R is the radius of the sun and T its temperature, then the energy emitted by the sun per sec through radiation in accordance with Stefan’s law will be given by P  eA T 4  4R 2T 4 U In reaching earth this energy will spread over a sphere of radius r (= average distance between sun and earth); so the intensity of solar radiation at the surface of earth (called solar constant S) will be given by P 4R 2T 4  4r 2 4r 2 ST S i.e.  r  2 S  T      R    1/4  1. 5  10 8    7  10 5  r 2  1. 4  10 3    5. 67  10 8  As r  1.5  10 8 km, R  7  10 5 km, S  2     1/4 ~ 5800 K R Sun Eart h cal kW W  1.4 2 and   5.67  10 8 2 4 2 m K cm min m This result is in good agreement with the experimental value of temperature of sun, i.e., 6000 K. The difference in the two values is attributed to the fact that sun is not a perfectly black body.