213.1 Thermal Conductivity At the end of this topic, students should be able to: Define heat as energy transfer due to temperature difference.Explain the physical meaning of thermal conductivity.Use rate of heat transfer,Use temperature-distance graphs to explain heat conduction through insulated and non-insulated rods, and combination of rods in series.
3Heat is defined as the energy that is transferred from a body at a higher temperature to one at a lower temperature , byconduction, convection or radiation.Heat always transferred from a hot region (highertemperature) to a cool region (lower temperature) untilthermal equilibrium is achieved.Heat is transferred by three mechanisms,1) Conduction2) Convection3) RadiationThermal Conduction is defined as the process wherebyheat is transferred through a substance from a regionof high temperature to a region of lower temperature.
4The mechanism of heat conduction through solid material (for extra knowledge only) BSuppose a rod is heated at one end (A).Before the rod being heated all the molecules vibrateabout their equilibrium position.As the rod is heated the molecules at the hot end (A)vibrate with increasing amplitude, thus the kinetic energyincreases.While vibrating the hot molecules collide with theneighbouring colder molecules result in transfer of kineticenergy to the colder molecules.This transfer of energy will continue until the cold end (B)of the rod become hot.
5Thermal conductivity, k Consider a uniform cylinder conductor of length l withtemperature T1 at one end and T2 at the other end asshown in figure above.The heat flows to the right because T1 is greater than T2.
6The rate of heat flow ,through the conductor is given by:
7The rate of of heat flow through an object depends on : Thermal conductivity.Cross-sectional area through which the heat flow.Thickness of the material.Temperature difference between the two sides of the material.
8The negative sign because the temperature T, become less as the distance, x increases.The rate of heat flow is a scalar quantityand its unit is J s-1 or Watt (W).Thermal conductivity , k is defined as the rate of heat flows perpendicularly through unit cross sectional area of a solid , per unit temperature gradient along the direction of heat flow.Thermal conductivity is a property of conducting material.( the ability of the material to conduct heat) where goodconductors will have higher values of k compared to poorconductors.
9Materials with large k are called conductors; those with small k are called insulators.
10Heat conduction through insulated rod Consider heat conduction through an insulated rod whichhas cross sectional area A and length x as shown above.If the rod is completely covered with a good insulator, noheat loss from the sides of the rod.By assuming no heat is lost to the surroundings, thereforeheat can only flow through the cross sectional area fromhigher temperature region, T1 to lower temperature region,T2 .
11The red lines (arrows) represent the direction of heat flow. insulatorThe red lines (arrows) represent the direction of heat flow.When the rod is in steady state (the temperature falls ata constant rate) thus the rate of heat flows is constant alongthe rod.This causes the temperature gradient will be constant alongthe rod as shown in figures above.
12Heat conduction through non-insulated rod The metal is not covered with an insulator, thus heat islost to the surroundings from the sides of the rod.The lines of heat flow are divergent and the temperaturefall faster near the hotter end than that near the colder end.
13Less heat is transferred to Y. This causes the temperature gradient graduallydecreases along the rod and result a curve graph wherethe temperature gradient at X higher than that at Y asshown in figure below.dTdxXYFromAnd from the graphwhere A and k are the same along the rod.ThusTemperature gradient , at any point on the rod isgiven by the slope of the tangent at that point.
14Combination 2 metals in series xcxDandinsulatorMaterial CMaterial D
15When steady state is achieved , the rate of heat flow through both materials is same (constant).From the equation of thermal conductivity, we getBut
16Example 13.1A metal cube have a side of 8 cm and thermal conductivity of 250 W m-1 K-1. If two opposite surfaces of the cube have the temperature of 90 C and 10 C, respectively. Calculatea) the temperature gradient in the metal cube.b) the quantity of heat flow through the cube in 10minutes.(Assume the heat flow is steady and no energy is lost to the surroundings)
17Solution 13.1A = l2 = (8x102)2 = 64 x104 m2, k = 250 W m-1 K-1, T1= 90C, T2 = 10Cb) Given t = 10 x 60 = 600 s
18Example 13.2 A 5 mm thick copper plate is sealed to a 10 mm thick aluminium plate and both have the same cross sectionalarea of 1 m2.The outside face of the copper plate is at100 C, while the outside face of the aluminium plate is at80 C.a) Find the temperature at the copper-aluminiuminterface.b) Calculate the rate of heat flow through the crosssectional area if heat flow is steady and no energy islost to the surroundings.(Use kCu = 400 W m-1C-1 and kAl = 200 W m-1C-1)
19Solution 13.2xCu= 5x103m, xAl= 10x103m, A= 1 m2, Tcu = 100C, TAl = 80CThe rate of heat flowthrough the copper and aluminium plate is same, therefore
21Solution 13.2xCu= 5x10-3m, xAl= 10x10-3m, A= 1 m2, Tcu = 100C, TAl = 80Cb) By applying theequation for rate of heatflow through the copperplate, hence
22Exercise1. A metal plate 5.0 cm thick has a cross sectional area of 300 cm2. One of its face is maintained at 100C by placing it in contact with steam and another face is maintained at 30C by placing it in contact with water flow. Determine the thermal conductivity of the metal plate if the rate of heat flow through the plate is 9 kW.(Assume the heat flow is steady and no energy is lost to the surroundings).( 214 W m-1K-1 )2. A rod m long consists of a m length of aluminium joined end to end to a m length of brass. The free end of the aluminium section is maintained at 150.0C and the free end of the brass piece is maintained at 20.0C. No heat is lost through the sides of the rod. At steady state, find the temperature of the point where the two metal are joined.(Use k of aluminium = 205 W m-1C-1 and k of brass = 109 W m-1C-1) (90.2C)
2313.2 Thermal expansion At the end of this topic, students should be able to: Define and use the coefficient of linear, area and volume thermal expansion.Deduce the relationship between the coefficients of expansion ,
2414.3 Thermal expansion Thermal expansion is defined as the change in dimensions of a body accompanying a change intemperature.3 types of thermal expansion :- Linear expansion- Area expansion- Volume expansionIn solid, all types of thermal expansion are occurred.In liquid and gas, only volume expansion is occurred.At the same temperature, the gas expands greater thanliquid and solid.
25Linear expansionConsider a thin rod of initial length, l0 at temperature,T0 is heated to a new uniform temperature, T and acquires length, l as shown in figure below.If ΔT is not too large (< 100o C)and
26Unit of is C-1 or K-1.The coefficient of linear expansion, is defined asthe change in length of a solid per unit length per unitrise change in temperature.If the length of the object at a temperature T is l,For many materials, every linear dimension changesaccording to both equations above. Thus, l could be thelength of a rod, the side length of a square plate or thediameter (radius) of a hole.
27For example, as a metal washer is heated, all dimensions including the radius of the hole increase as shown infigure below.
28Area expansionThis expansion involving the expansion of a surface area ofan object.Consider a plate with initial area, A0 at temperature T0 isheated to a new uniform temperature, T and expands by A,as shown in figure below.From this experiment, we getand
29Unit of is C-1 or K-1The coefficient of area expansion, is defined as thechange in area of a solid surface per unit area perunit rise in temperature.The area of the of the surface of object at atemperature T can be written as,For isotropic material (solid) , the area expansion isuniform in all direction, thus the relationship between and is given by
30DerivationConsider a square plate with side length, l0 is heated and expands uniformly as shown in figure below.becausewhereandcompare with
31Volume expansionConsider a metal cube with side length, l0 is heated and expands uniformly. From the experiment, we getandUnit of is C-1 or K-1.The coefficient of volume expansion, is defined as the change in volume of a solid per unit volume per unit rise in temperature.
32The volume of an object at a temperature T can be written as,For isotropic material (solid), the volume expansion isuniform in all direction, thus the relationship between and is given byDerivationConsider a metal cube with side length, l0 is heated and expands uniformly.lolΔl
34Example 13.3The length of metal rod is cm at 20C and cm at 45C, respectively. Calculate the coefficient of linear expansion for the rod.Solutionl0= cm, T0= 20C , l= cm, T = 45C
35Example 13.4A steel ball is cm in diameter at 20.0C. Given that the coefficient of linear expansion for steel is 1.2 x 105 C-1, calculate the diameter of the steel ball ata) 57.0Cb) 66.0CSolutiond0= cm, T0= 20.0C , = 1.2x105C-1a)b)
36Example 13.5A cylinder of radius 18.0 cm is to be inserted into a brass ring of radius 17.9 cm at 20.0C. Find the temperature of the brass ring so that the cylinder could be inserted.(Given the coefficient of area expansion for brass is 4.0 x 105 C-1)Solutionrc= 18.0 cm, r0b= 17.9 cm, T0= 20.0C , = 4.0x105C-1When the cylinder pass through the brass ring, thus
37Example 13.6Determine the change in volume of block of cast iron 5.0 cm x 10 cm x 6.0 cm, when the temperature changes from 15 oC to 47 oC. ( = oC -1 )Solution
38Exercise A rod 3.0 m long is found to have expanded 0.091 cm in length after a temperature rise of 60 o C. What is the coefficient of linear expansion for the material of the rod ?5.1 x 106 o C12. The length of a copper rod is m and the length ofa wolfram rod is m at the same temperature.Calculate the change in temperature so that the tworods have the same length where the final temperaturefor both rods is equal.(Given the coefficient of linear expansion for copper is1.7 x 105 C1 and the coefficient of linear expansion for wolfram is 0.43 x 105 C1)78.72C
39Thermal Expansion of Liquid in A Container When a liquid in a solid container is heated, both liquid andthe solid container expand in volume.Liquid expands more than the solid container.The change in volume of a liquid is given byThe coefficient of volume expansion of a liquid is defined inthe same way as the coefficient of volume of a solid i.e :
40Example 13.7A glass flask with a volume of 200 cm3 is filled to the brim with mercury at 20 oC. How much mercury overflows when the temperature of the system is raised to 100 oC ?Solution