Thermal Conductivity

6- Thermal Conductivity Abstract: We will be measuring the thermal conductivity of an insulator plate using a method called “Lee’s Disc”. This method is used to find the thermal conductivity of bad conductors. Introduction: Heat is defined as the transfer of energy between two systems brought into thermal contact. In an atomic scale, this process is seen as the transfer of kinetic energy between particles. This is due to adjacent atoms vibrating against one another, or electrons moving from one atom to another. When a system is heated its particles gain kinetic energy, so they vibrate with higher amplitudes. Some materials, such as metals, are better conductors of heat that others. That’s mainly due to metals having a large number of electrons which aids the transfer of energy. Other materials are thought of as insulators, this means that they transfer heat poorly. Solids in general are better conductors than liquids and gases due to molecules being closer to one another resulting in more possible collisions and electron transfer. Polymers are usually insulators, but a team of researchers in MIT have recently been able to transform polyethylene into a heat conductor in one direction while maintaining an electrical insulator. Temperature is related to the average kinetic energy of the particles and is measured using a thermometer. In theory, the lowest temperature, reflecting no kinetic energy of particles, is called the absolute zero and is 0 kelvin. Conduction of heat occurs when two systems with different temperatures are in thermal contact. Heat is transformed from the system with higher temperature to the system with lower temperature. If we had a slab of a material with thickness ∆𝑥 and a cross-sectional area 𝐴 separating two systems with temperatures 𝑇! and 𝑇" where 𝑇" > 𝑇! , we denote the rate of change of heat as ∆" ∆# where the rate of change of heat is directly proportional to the cross-sectional area 𝐴 and the difference in temperatures between the two systems ∆𝑇 and inversely proportional to the thickness of the slab ∆𝑥. ∆𝑄 ∆𝑇 ∝𝐴 ∆𝑡 ∆𝑥 If we take a slab of infinitesimal thickness 𝑑𝑥 and systems with infinitesimal temperature difference 𝑑𝑇, we can rewrite as: ∆𝑄 𝑑𝑇 = −𝑘𝐴 ∆𝑡 𝑑𝑥 where 𝑘 is a constant denoting the thermal conductivity of the material and the minus sign denotes that the transfer occurs from the higher temperature to the lower. It’s convenient to denote the rate of change of heat as 𝐻. 𝐻 has units of watts (𝑄 is in joules). So, we reach the finals form 𝐻 = −𝑘𝐴 #$ #% 𝑑𝑇 𝑑𝑥 is called the temperature gradient, or the variation of temperature with position. When the temperatures of the two systems become constant, we reach a state called “Steady state”. In this case the temperature along the thickness of #$ the glass plate is constant and the quantity #% becomes: 𝑑𝑇 𝑇! − 𝑇" = 𝑑𝑥 ∆𝑥 And thus, the heat transfer 𝐻: 𝑇1 − 𝑇2 𝑇2 − 𝑇1 𝐻 = −𝑘𝐴 = 𝑘𝐴 ∆𝑥 ∆𝑥 We also know that the amount of heat absorbed can be written as: 𝑄 = 𝑚𝑐∆𝑇 Or 𝑑𝑄 = 𝑚𝑐 𝑑𝑇 𝑑𝑄 𝑑𝑇 = 𝑚𝑐 =𝐻 𝑑𝑡 𝑑𝑡 When 2 systems are in thermal contact, the heat absorbed by the system in lower temperature must be equal to the amount of heat lost by the other higher temperature system. By equating the two expressions for 𝐻 as heat absorbed and released, we can say that at the steady state temperature: 𝑑𝑇 𝑇1 − 𝑇2 𝑚𝑐 𝑑𝑡 = 𝑘𝐴 ∆𝑥 We can then rearrange and get an expression for 𝑘: 𝑘= 𝑚𝑐∆𝑥 𝑑𝑇 (𝑇$ − 𝑇% )𝐴 𝑑𝑡 If we graph the T-t diagram, we will get a nonlinear curve. The slope of the tangent line at 𝑇 = 𝑇! will be used as &' &# . Procedure: 1- Set the experiment as shown in a previous diagram. 2- Once the temperature readings on the thermometer become constant, take the readings of 𝑇" and 𝑇! . Those are the steady state temperatures. 3- Remove the glass plate that is between the two discs and let the two discs be in contact. This will raise the temperature of the disc to above the steady state. 4- Record the temperature of the lower disc then remove the steam chamber and the first disc and cover the lower disc with an insulator. 5- Record the temperature of the second disk every 30 seconds after covering it. 6- Draw the T-t diagram. At 𝑇 = 𝑇! take the tangent line’s slope and use the previous equation to calculate 𝑘. Data: t(s) T(℃) 0 30 60 90 120 150 180 210 240 270 300 92 90.6 89.5 88.6 87.9 87.2 86.6 86.1 85.7 85.4 85.1 𝑇! = 87 ℃ 𝑇" = 91 ℃ 𝑚 = 940 𝑔 𝑑 = 15 𝑐𝑚 𝑐 = 360 𝐽⁄𝑘𝑔 ∙ ℃ ∆𝑥 = 3𝑚𝑚 𝑠𝑙𝑜𝑝𝑒 = 𝑘= ∆𝑇 = −0.0208 ℃9𝑠 ∆𝑡 𝑚𝑐∆𝑥 (𝑠𝑙𝑜𝑝𝑒) = 0.298 𝑊9 𝑚∙℃ (𝑇2 − 𝑇1 )𝐴 Error: " ∆𝑚 " ∆(∆𝑥 ) ∆𝑠𝑙𝑜𝑝𝑒 " ∆𝑇 " ∆𝐴 " ∆𝑘 = 𝑘 <= ? + A B += ? += ? += ? 𝑚 ∆𝑥 𝑠𝑙𝑜𝑝𝑒 𝑇 𝐴 Where: 2∆𝑑 ! ) 𝑑 ∆𝐴 = 𝐴$( ∆𝑇 ! ∆𝑡 ) + ( )! 𝑇 𝑡 ∆𝑠𝑙𝑜𝑝𝑒 = 𝑠𝑙𝑜𝑝𝑒$( Sources of error: 1- Accuracy in measurement tools. 2- Human error. Not reading the measurement accurately. 3- Tangent line not accurate. 4- Poor insulation of the system. 5- Approximation. Discussion and Conclusion: We were able to find the Thermal conductivity 𝑘 of the insulator. The conclusions we reach: 1- Heat transfers between systems when there is a difference in temperatures. 2- The rate of flow of heat is directly proportional to the cross-sectional area 𝐴 and the difference in temperatures between the two systems ∆𝑇 and inversely proportional to the thickness of the slab ∆𝑥. ∆) #$ 3- The minus sign in ∆* = −𝑘𝐴 #% indicates that heat transfers from higher temperatures to lower temperatures. 4- The thicker the slab is, the slower it will transfer heat. 5- The larger the temperature difference is, the faster heat will transfer. 6- The absorbed energy by the system with lower temperature must equal the lost temperature by the higher temperature system. 7- The higher the value of 𝑘, The better conductor the material is. 8- After enough time the systems temperature will stop changing, this is known as steady state, or equilibrium.