Handout 10: Heat and heat transfer. Heat capacity

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1 1 Handout 10: Heat and heat transfer Heat capacity Consider an experiment in Figure 1. Heater is inserted into a solid substance of mass m and the temperature rise T degrees Celsius is measured by a thermometer. The container is well insulated so that little or no heat is lost. If amount of heat Q is provided by the heater, it is found that Q m T Q = mc T. The constant c is known as the specific heat capacity which is a property of substance. Heat capacity C of the substance is defined as C = mc. Figure 2 shows specific heat capacity of some substances. The values of c of solids are much less than those of liquids and gases. Figure 1: Experiment of finding heat capacity of a substance If a substance absorbs heat, its temperature increases. If the substance releases heat, its temperature drops. The rise or fall in temperature is measured in degrees Celsius T C. However, it can also be measured in kelvins T K, T K = T C + 273, because T in degrees Celsius is the same as T in kelvins. Latent heat Figure 2: Specific heats of some materials Not only can heat change the temperature of a substance, heat can also change phases of the substance. Figure 3 shows variation of temperature of substance with the heat input. The phase of the substance changes from solid to liquid and to gas. It should be noted that during phase change, the temperature is constant. The amount of heat required to change the phase is called latent heat. The heat used to change phase from solid to liquid at freezing point is called latent heat of fusion. The heat required to change phase from liquid to gas at boiling point is called latent heat of vaporization. Consider a substance of mass m, the latent heats are given by Figure 3: Variation of temperature and energy input Q = ml f, Q = ml v,

2 where L f and L v are known as specific latent heat for fusion and specific latent heat for vaporization respectively. For water at atmospheric pressure, L v = J kg -1 and L f = J kg -1. Example 1 Energy is supplied at constant rate 1800 Js -1 to 1-liter water to raise temperature from 20 C to boiling point. How long will it take the water to boil? 2 Example 2 The graph shows variation of temperature with amount of heat absorbed by 1-kg substance. a) Find specific heat capacity c for solid, liquid and gaseous phase of this substance. b) Find L v and L f. Example 3 Water of mass 2 kg at temperature 25 C is mixed with water of mass 3 kg at temperature 45 C. Determine the final temperature of the mixture. Assume that no heat is lost. Example 4 How much heat is required to turn solid ice of mass 0.50 kg at temperature 5 C into vapor at 100 C? The specific heat capacity of solid ice is 2000 J kg -1 K -1.

3 3 Thermal expansion Most solid materials expand if the temperature increases. At high temperatures, atoms of solid vibrate more violently, pushing the neighboring atoms. Therefore the spacing between atoms increases, making the solid expands. In Figure 4, suppose a rod of length L 0 is heated so that the rise in temperature is T. The result is that the length of the rod extends by L. Linear thermal expansion means that L is proportional to L 0 and T: Figure 4: Linear thermal expansion of a rod L = αl 0 T where α is called the coefficient of linear expansion which depends on types of materials. The length of the rod is therefore given by L L 0 = αl 0 T L = L α T. In construction, thermal expansion has to be accounted for, otherwise damage could occur. The gaps are often left at the joint of concrete roads (Figure 5) to give room for thermal expansion of the roads. Example 5 A steel rod of length 2.55 m at temperature 20 C is heated. How much does the rod extends when the temperature is 150 C? Steel has the coefficient of linear expansion α = C 1. Figure 5: Joint in the road bridge to avoid damage from thermal expansion Example 6 A solid has coefficient of linear expansion α. Show that the coefficient of area expansion is 2α

4 4 Heat transfer 1. Conduction Heat conduction is a flow of heat in solid between two ends with a temperature difference. Heat flows from hotter end to the cooler end. The solid medium does not move during the heat conduction. Consider a setup in Figure 6. A bar of length d and cross-section area A is has one end at temperature T 2 and the other at temperature T 1 T 2 > T 1. At steady state, the rate of heat transfer is constant and is given by Figure 6: Heat conduction between two ends with temperature difference T 2 T 1. dq dt = P = κa T d, where κ is the thermal conductivity of the rod. The values of thermal conductivity of common materials are shown in Figure 7. Note that the unit of P is watt (W). 2. Convection Heat convection is the transfer of heat due to the movement of warm fluid flowing to displace the cooer fluid. An example of heat convection is the formation of breezes as shown in Figure 8. During the day time, the land is warmer and the warm air above the land rises. Cooler air from the sea replaces the risen hot air, causing sea breeze. At night, the sea is warmer and the air rises. The cooler air from the land replaces, resulting in land breeze. Figure 7: Thermal conductivity of common materials 3. Radiation Thermal radiation is a transport of energy in form of electromagnetic waves. This means that thermal radiation can occur in vacuum. An object with surface area A and absolute temperature T radiates energy at rate, according to Stefan-Boltzmann s law, dq dt = P = εσat4, where ε 0 < ε 1 is emissivity and σ is called the Stefan-Boltzmann constant, σ = Wm -2 K -4. Black body radiates energy perfectly with ε = 1. Radiation from the sun is a good approximation of the black-body radiation. Figure 8: Sea breeze during the day and land breeze at night as the result of heat convection

5 Example 7 Both ends of an aluminium rod of radius r = 1.5 cm and length 0.95 m have temperature difference of 110 C. Evaluate the rate at which heat is transferred in the rod. 5 Example 8 Steel rod and copper rods with the same cross-section area are joined. The ratio of the length of steel rod to that of copper rod is 1:2. If the free end of the steel rod is kept at 100 C and the free end of the copper rod is at 0 C, determine the temperature T at the junction. Example 9 Determine the rate of energy radiation at the surface of the sun given that the surface temperature of the sun is about 6000 K and the radius of the sun is m.