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Thermodynamics, heat transfer, conduction equation
- 1. By Abhay L 1. Thermodynamicsand heat transfer 2. Heat conduction equation in Cartesian coordinates
- 2. Content 1. Introduction 2. Lawsof thermodynamics 3. Mode of heat transfer 4. General heat conduction equation for constant thermal conductivity
- 3. 1. Introduction Temperature: • Indicatesthe thermal state of a system or a body. • Temperature is a measure of internal energy possessed by a system Heat and work: • Form of energy which are transient in nature • Heat-Flows from high temperature to low temperature. • Work_ Force and displacement Thermodynamics Heat Transfer Thermodynamics is a science which deals with amount of energy possess by the system Heat transfer is a science that deals with the rate as well as mode of heat transfer during a process. Thermodynamics is a science which deals with amount of energy transfer during the process and their relation with properties of the system Heat transfer indicates the temperature distribution inside a body.
- 4. 2. Laws ofThermodynamics Zeroth law of thermodynamics • Thermal equilibrium concept • ΔU = Q - W First law of thermodynamics • Heat and wok are mutually convertible • heat flows spontaneously from a hot object to a cold object • Close cycle: σ 𝑄 = σ 𝑊 • For a process δ𝑄 = 𝑑𝑈 + δ𝑊 Second law of thermodynamics • Limited amount of work can be obtained from given quantity of heat energy • Claussius statement • Kelvin-Planck statement Third law of thermodynamics The entropy of a system approaches a constant value as its temperature approaches absolute zero. Limiting the behavior of system.
- 5. 3. Mode ofheat transfer Conduction • Medium required • Solid: Lattice vibration and Transport of free electrons • Liquid and gas: Collision and diffusion • Bulk motion of molecules is zero • k- Thermal conductivity, W/m-K Convection • Liquid and gas • Bulk motion of fluid molecules is low velocity - natural convection (buoyancy force) • Bulk motion of molecules with high velocity- Force convection (pump or blower) • Medium required • H: heat transfer coefficient, W/m2-K Radiation • Electromagnetic waves emitted by atomic and subatomic agitation at the surface • No medium required, can takes place in vacuum • Boltzmann constant, 𝜎: 5.67× 10-8 W/m2-K4 • 𝜀= Emissivity and 0 ≤ 𝜀 ≤ 1 Newton’s law of cooling, 𝑄 = ℎ𝐴 (𝑇2 − 𝑇1) Stefan's Boltzmann law, 𝑄 = 𝜀𝜎𝐴 (𝑇2 4 − 𝑇1 4 ) Image source: https://www.baamboozle.com/game/38934 Fourier law, 𝑄 = 𝑘𝐴 𝑇2−𝑇1 𝐿
- 6. 4. General Heatconduction equation in Cartesian coordinates Fourier law, 𝑄𝑥 = −𝑘𝐴𝑥 𝑑𝑇 𝑑𝑥 𝑄𝑥+𝑑𝑥 Y X Z 𝑄𝑥 dx dz dy Net heat stored due to conduction in x direction = 𝑄𝑥−𝑄𝑥+𝑑𝑥 = 𝑄𝑥−(𝑄𝑥+ 𝜕𝑄𝑥 𝜕𝑥 𝑑𝑥) = − 𝜕 𝜕𝑥 (𝑘𝑥𝑑𝑦 𝑑𝑧 𝜕𝑇 𝜕𝑥 )𝑑𝑥 = 𝜕 𝜕𝑥 (𝑘𝑥 𝜕𝑇 𝜕𝑥 )𝑑𝑉 …………… (1) = 𝜕 𝜕𝑦 (𝑘𝑦 𝜕𝑇 𝜕𝑦 )𝑑𝑉 …………… (2) = 𝜕 𝜕𝑧 (𝑘𝑧 𝜕𝑇 𝜕𝑧 )𝑑V …………… (3) Internal energy stored in the body 𝑄𝑖𝑛𝑡= 𝑑𝑈 𝑑𝑡 = 𝑑 𝑑𝑡 𝑚 . 𝐶. 𝑑𝑇 = 𝑚 . 𝐶. 𝑑𝑇 𝑑𝑡 = 𝜌. 𝑑𝑉 . 𝐶. 𝑑𝑇 𝑑𝑡 …. (6) Chemical reaction or current pass through it 𝑄𝑔𝑒𝑛= 𝑞𝑣. 𝑑𝑉 ….(5) where 𝑞𝑣 = 𝑟𝑎𝑡𝑒 𝑜𝑓 ℎ𝑒𝑎𝑡 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝑊 𝑚3 Law of conservation of energy; (1)+(2)+(3)+(5)= (4) 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 + 𝑞𝑣= 𝜌𝑐 𝑑𝑇 𝑑𝑡
- 7. 5. General Heatconduction equation for constant thermal conductivity Fourier-Biot Equation 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 + 𝑞𝑣= 𝜌𝑐 𝑑𝑇 𝑑𝑡 For isotropic material: 𝑘𝑥 = 𝑘𝑦 = 𝑘𝑧 = 𝑘 𝑘 𝜕𝑇2 𝜕𝑥2 + 𝜕𝑇2 𝜕𝑦2 + 𝜕𝑇2 𝜕𝑧2 + 𝑞𝑣= 𝜌𝑐 𝑑𝑇 𝑑𝑡 In vector form 𝑘 𝛻. 𝛻𝑇 + 𝑞𝑣= 𝜌𝑐 𝑑𝑇 𝑑𝑡 𝛻2𝑇 + 𝑞𝑣 𝑘 = 𝜌.𝑐 𝑘 . 𝑑𝑇 𝑑𝑡 𝜗 = 𝜇 𝜌 𝑚2 𝑠 𝑘𝑖𝑛𝑒𝑚𝑎𝑡𝑖𝑐 𝑣𝑖𝑠𝑐𝑜𝑐𝑖𝑡𝑦 − 𝑖𝑛 𝑓𝑙𝑢𝑖𝑑 𝛼 = 𝑘 𝜌.𝑐 --- 𝑊 𝑚.𝐾 ∗ 𝑚3 𝑘𝑔 ∗ 𝑘𝑔.𝐾 𝐽 = 𝑚2 𝑠 𝛼 = 𝑇ℎ𝑒𝑟𝑚𝑎𝑙 𝑑𝑖𝑓𝑖𝑠𝑖𝑣𝑖𝑡𝑦 − 𝑖𝑛 ℎ𝑒𝑎𝑡 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟 1 𝑟𝑛 𝜕 𝜕𝑟 𝑟𝑛 𝜕𝑇 𝜕𝑟 + 𝑞𝑣 𝑘 = 1 𝛼 . 𝑑𝑇 𝑑𝑡 One dimension heat conduction equation Slab, n=0 Cylinder, n=1 Sphere, n=2
- 8. 4. General Heatconduction equation No internal heat generation (Diffusion equation) 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 = 𝜌𝑐𝑝 𝜕𝑇 𝜕𝑡 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 = 0 Steady state and no heat generation (Laplace equation) 𝜕 𝜕𝑥 𝑘 𝜕𝑇 𝜕𝑥 + 𝜕 𝜕𝑦 𝑘 𝜕𝑇 𝜕𝑦 + 𝜕 𝜕𝑧 𝑘 𝜕𝑇 𝜕𝑧 + 𝑞𝑣= 0 Steady state (Poisson’s equation) One dimension without heat generation and steady state 𝜕𝑇2 𝜕𝑥2 = 0 ; 𝜕𝑇 𝜕𝑥 = 𝑐1; 𝑇 = 𝑐1𝑥 + 𝑐2 𝑇1 𝑇2 𝑄 𝑥0 𝑥𝐿 𝑥 = 0; 𝑇1 = 𝑐2 𝑥 = 𝐿; 𝑇2 = 𝑐1. 𝐿 + 𝑇1 ; 𝑐1 = 𝑇2−𝑇1 𝐿 𝑇 = 𝑇2−𝑇1 𝐿 𝑥 + 𝑇1
- 9.