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Thermals 1 Summary - Thermal Fluids 1 Summary Sheet Formula...

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May 8, 2009 Thermal & Fluids 1 Summary Sheet Carl Hansen Types of Energy : Static Energies: Formula Dynamic Energies Formula Kinetic Energy KE = ½m* v 2 Work w = ∫ F∙dx w = ∫ P*d V Gravitational Potential Energy GPE = m*g*h Heat Transfer Q = m*C*ΔT Strain Energy SPE = ½k*x 2 Electrical power dw/dt = i*V Internal Thermal Energy ΔU = m*C*ΔT Chemical & Nuclear Energy 1 st law of Thermodynamics: ΔE = Q – w “increase in system energy = heat added – work done by system.” ΔE = Δ(U + PE + KE) Model: heat engine system Gases: Ideal Gas law: P*V = m* K M R T * ° ; where M is molar mass, R = 8.31434 kJ/(kmol * °K) = 0.08205 K * mol atm * L ° Expansion Work: w = ∫ P*d V M air = 28.95 kJ/(kmol*°K) Special Cases: Constant Volume (d V /dt = 0): w = 0 Constant Pressure (dP/dt = 0): w = P( V 1 V 2 ) Isothermal (dT/dt = 0, or P* V =const): w = P 1 * V 1 * ( 29 1 2 V V ln Polytrophic (P* V n =const): Adiabatic(no heat transfer): n 1 V * P V * P w 1 1 2 2 - - = n = k = C p /C v k air = 1.4 Specific Heats of ideal gases: C p : if it has constant pressure C v : if it has constant volume Relation: Cp = Cv + M R Heat Transfer : General idea: use electrical analogy R T T Q cold hot - = How to calculate thermal resistances: Conduction Convection Radiation A * k L R = A * h 1 R = ( 29 4 g surroundin 4 T T * A * * Q - σ ε = 3 2 T T g surroundin * 4 * A * * 1 R σ ε + Series: (such as layers of a wall) Parallel: (such as 4 different walls, a ceiling and a floor) = i R R = i R 1 1 R Conduction for extended common geometries: Slab Cylindrical Pipe Spherical Shell A * k L R = ( 29 k * L * 2 ln R 1 2 r r π = k * r * r * 4 r r R 2 1 1 2 π - = Infinite sphere: R= 1 r * k * 4 1 π Generalization (Layers of insulation + convection) n n n 1 n 1 1 r * 2 * L * h 1 L * k * 2 ) r / r ln( r * 2 * L * h 1 R π π π + + = + z A r 1 r 2 L
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May 8, 2009 Thermal & Fluids 1 Summary Sheet Carl Hansen Lumped Systems: A system in which we assume constant temperature throughout, and take an interest in the temperature vs. time function. Criteria: R cond < < R conv Test the Biot Number: Bi ≡ h*L char /k L char V /A if Bi < ≈0.1, it is a lumped system. Problem set-up: dQ/dt = d[m*C*ΔT]/dt = convection g surroundin R T T Q - = m*C* R T T s dt dT - = Soultion: g surroundin t * C * m A * h g surroundin 0 T e * ) T T ( ) t ( T + - = - Time constant: A * h C * m τ
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