Lecture 12 heat transfer.
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Heat can be transferred through three mechanisms: conduction, convection, and radiation. Conduction involves the transfer of heat between objects in direct contact through collisions of molecules. Convection involves the transfer of heat by the movement of fluids like gases and liquids. Radiation involves the emission and absorption of electromagnetic waves and can occur through a vacuum. The rate of heat transfer by conduction follows Fourier's Law and depends on factors like thermal conductivity, area, and temperature difference. Materials with high thermal conductivity like metals are good conductors while materials with low conductivity like wood and air are good insulators. Radiation transfer follows the Stefan-Boltzmann law and depends on emissivity, area, and the temperature difference between objects.
Lecture 12 heat transfer.
- 2. Mechanisms of heat transfer Conduction: Two surfaces in contact, or within a body. Convection: Particles move between regions of different temperature and carry energy Radiation: Energy carried by electromagnetic waves. Can happen through vacuum!
- 3. Conduction Metal bar of length L and cross-sectional area A between two sources at THOT and TCOLD. TH −TC dQ H = = kA dt L k = thermal conductivity This formula “makes sense”: What transfers energy are the collisions between faster and slower molecules, and the number of collisions increases with cross-section area, A. Also, the rate of transfer of energy decreases with length. Thermal conductivity is the efficiency of collisions to exchange energy
- 4. Thermal conductors and insulators Of course, there are no perfect thermal conductors or insulators. Conductors (high k) Metals 50-500 W/(m K) Insulators (low k) Wood Air (still) Styrofoam 0.12-0.04 W/(m K) 0.024 W/(m K) 0.01 W/(m K) How low can you go?????…. “Aerogel”: 8x10-5 W/(m K) What’s aerogel…. • • • a man-made substance formed by specially drying a wet silica gel, resulting in a solid mesh of microscopic strands. used on space missions to catch comet dust the lightest material know to man, according to the Guiness Book of World Records -- really, really light. It is ~98% porous, and yet it is quite rigid…
- 5. Aerogel 2.5 kg brick 2 g aerogel DEMO: Water is also a poor conductor k = 8 x 10-5 W/m-K
- 6. Example: Window If it’s 22°C inside and 0°C outside, what is the heat flow through a glass window of area 0.3 m2 and thickness 0.5 cm? The thermal conductivity of glass is about 1 W/m-K. W 0.3m2 kA H = ∆T = 1 ÷( 22K ) = 1320 W ÷ L mK 5 × 10 −3 m ÷
- 7. Example: Double pane How much heat is lost (per second) through a double-pane version of that window, with a 0.5-cm air gap? The thermal conductivity of air is about 0.03 W/(m K). We can pretty safely ignore the glass, which has a much higher conductivity than air. H is limited by conduction across the air gap. The heat current is limited by the air gap. W 0.3m2 H = 0.03 ÷( 22K ÷ mK 5 × 10 −3 m ÷ ) = 39.6 W (<< 1320 W) Note: larger air gaps don’t always work better because convection currents swirl the hot and cold air around.
- 8. ACT: Hot cup A closed container that is not a good heat insulator contains water at 55°C. It is in a room where the temperature is 25°C. Which of the following graphs shows the temperature of the water vs. time? T T T 55 55 55 25 25 25 time A B time C time
- 9. Consider hot cup of water, TH, sitting in cold room TC – assume room is large enough so that TC is fixed – heat transferred from water to room at rate dQ/dt TH −TC dQ A = surface area of cup = kA L = thickness of cup walls dt L What happens next ? ⇒ water has lost energy ⇒ TH is reduced, closer to TC ⇒ lower rate of heat transfer dQ/dt ⇒ This lowers the rate at which TH is reduced T To solve find the temperature as a function of time we need a differential equation… 55 25 time
- 10. Heat flow vs current flow Heat flow: kA H = ∆T L Definition: Thermal resistance Rth ≡ 1 L kA ∆T H = Rth Compare to Electric current (charge flow): L 1 L R ≡ρ = A κA ∆V I = R
- 11. How long does heat conduction take? hot General problem: TA A cA Isolated system: H k TB cold B cB Heat released by A = Heat absorbed by B Heat exchange through bar with thermal conductivity k How long until thermal equilibrium?
- 12. To simplify, let us assume that one of the objects (A) is really large (ie, constant temperature). dQ = mBcBdT = C BdT The temperature in B changes. Every change in temperature is caused by some heat being transferred: At time t, the temperature of B is T. The flow rate: (C = mc thermal capacity ) T −T dQ kA = TA −T ) = A ( dt L Rth dQ dQ dT dT = = CB dt dT dt dt CB dT TA −T = dt Rth dT 1 = dt TA −T C BRth T −T t ln A =− TA −TB τ T TA τ = C BRth T −TA = (TB −TA ) e t − τ TB Relate to: Charging/discharging a capacitor in RC circuit. t
- 13. Convection Particles move between regions of different temperature. Mathematically… waaaay beyond 222 Example: Wind chill
- 14. DEMO: Lighting a match Radiation All pieces of matter emit electromagnetic radiation – typically long-wavelength, infrared – For hot bodies, radiation is in visible wavelengths e.g. light-bulbs Rate of radiation of heat for a body a temperature T: Stefan-Boltzmann Law dQ = −Ae σT dt 4 A : area of surface of body e : emissivity of surface, empirical (e.g e = 0.3 for Cu) σ: Stephan-Boltzmann constant J T in Kelvins σ = 5.67 × 10 −8 m2 ×s × 4 K Radiation is in all wavelengths. Depending on T, most heavily in one wavelength interval than other. This is NOT described by this law
- 15. ACT: Radiating book A book at room temperature in this room radiates energy. Yet the temperature of the book does not go down: why not? A. Radiation stops when book is at room temperature B. Heat is also absorbed by the book as radiation from the room C. There must be a heat-source in the book dQradiated = −Abookebookσ ( book ) 4 T dt dQabsorbed = +Abookebookσ ( room ) 4 T dt if Troom =Tbook ⇒ dQradiated dQabsorbed + =0 dt dt ⇒ temperature unchanged
- 16. Emission and absortion The term radiation refers to emission (dQ/dt < 0) and absortion (dQ/dt < 0) of energy: dQ dt ( = −Ae σ T net 4 4 −Tenvironment ) Emissivity: e=1 black body: ideal emitter and ideal absorber (absorbs all the radiation that strikes it) eg. Sun e=0 ideal reflector (absorbs no radiation and does not emit either) eg. Mirrored lining inside thermos bottles
- 17. In-class example: Sitting inside a fridge You are sitting in your shorts inside a cold (0°C) room. At what rate are you transferring heat to the room by radiation? (Take the worst case scenario with e = 1 and A = 2 m2) Flow from body to room A. 0 B. 2.4×10-4 W C. 404 W D. 8.9 ×104 W E. None of the above dQ dt ( = Ae σ T net 4 4 −Tenvironment ) J 4 4 4 = 2 m2 5.67 × 10 −8 2 ÷ 309 − 273 K m ×s × 4 K = 404 W ( ) Note: In just 10 s, this is a loss of 4000 J ~ 1000 cal = 1 Cal (food calorie). We burn food to stay warm! ( )