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1 HEAT TRANSFER PROBLEMS PhysicsPhysics EnvironmentalEnvironmental Equipo docente: Alfonso Calera Belmonte Antonio J. Barbero Departamento de Física Aplicada UCLM

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2 PhysicsPhysics EnvironmentalEnvironmental PROBLEM 1. The differential equation giving the profile of temperature through a hollow cylinder infinitely long at steady state is Where r is the radious Find out the heat flux per unit of length in a cylindrical pipe with the boundary conditions T=T 1 at r = r 1, T=T 2 at r=r 2 (the border effects are neglected, and it is assumed that conductivity k is constant). From boundary conditions

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3 PhysicsPhysics EnvironmentalEnvironmental PROBLEM 1 (continued) Heat fluxFrom Fourier’s equation Here A r means the curved surface of a cylindrical area, given by r L Note that

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4 PhysicsPhysics EnvironmentalEnvironmental PROBLEM 2. A metallic cylindrical pipe (thermal conductivity k m ) having an inner radious r 1 and an outer radious r 2 is covered by a d-cm thickness insulation of thermal conductivity k i. The pipe carries a fluid at T 1 temperature, and the external temperature is T’. Calculate the losses of heat per meter of isolated pipe. d r1r1 r2r2 r3r3 kmkm kiki T’ = T 3 From the solution of the prior problem: we use the concept of thermal resistence The heat flux is given by a set of terms having each the form Thermal resistences are given in this case by terms of the form Numerical application: k m = 40 W/mºC, k i = 0.75 W/mºC, r 1 = 5 cm, r 2 = 7 cm, d = 3 cm, T 1 = 80 ºC, T’ = 25 ºC. Find out the intermediate temperature T 2. T1T1 T2T2 T3T3

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5 PROBLEM 3. A metallic cylindrical pipe (thermal conductivity k m = 40 W/mºC) having an inner radious r 1 = 5 cm and an outer radious r 2 = 7 cm carries a fluid at T 1 = 80 ºC. The inner and outer heat transfer coefficients are 2500 W m -2 ºC and 1600 W m -2 ºC. The external temperature is 25 ºC. Find out the losses of heat per meter of pipe. PROBLEM 4. Consider three panes of glass, each of thickness 5 mm. The panes trap two 2.5 cm layers of air in a large glass door. How much power leaks through a 2.0 m 2 glass door if the temperature outside is -40 ºC and the temperature inside is 20 ºC? Data: k glass = 0.84 W/mºC, k air = 0.0234 W/m ºC (discuss if we can consider air in the same way we consider glass). PROBLEM 5. If the temperature of the Sun fell 5%, and the radius shrank 10%, what would be the percentage change of the Sun’s power output? PROBLEM 6. We know that the sun radiates 3.74·10 26 W. We also know that the distance from Sun to Earth is 1.5·10 11 m and the radius of Earth is 6.36·10 6 m. What is the intensity (power/m 2 ) of sunlight when it reaches Earth? How much power is absorbed by Earth in sunlight? (assume that none of the sunlight is reflected) What average temperature would allow Earth to radiate an amount of power equal to the amount of sun power absorbed?

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6 PROBLEM 5. If the temperature of the Sun fell 5%, and the radius shrank 10%, what would be the percentage change of the Sun’s power output? The power output would decrease 34% PROBLEM 6. We know that the sun radiates 3.74·10 26 W. We also know that the distance from Sun to Earth is 1.5·10 11 m and the radius of Earth is 6.36·10 6 m. What is the intensity (power/m 2 ) of sunlight when it reaches Earth? How much power is absorbed by Earth in sunlight? (assume that none of the sunlight is reflected) What average temperature would allow Earth to radiate an amount of power equal to the amount of sun power absorbed?

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7 From the radiated powerwe get the power of sunlight at the Earth’s orbit From the Sun, the Earth appears as a tiny disk whose radious is 6.36·10 6 m The power intercepted by Earth is If Earth would behave as a perfect blackbody 5.67051 x 10 -8 W/m 2 K 4 T Earth = 276 ºK PROBLEM 6. We know that the sun radiates 3.74·10 26 W. We also know that the distance from Sun to Earth is 1.5·10 11 m and the radius of Earth is 6.36·10 6 m. What is the intensity (power/m 2 ) of sunlight when it reaches Earth? How much power is absorbed by Earth in sunlight? (assume that none of the sunlight is reflected) What average temperature would allow Earth to radiate an amount of power equal to the amount of sun power absorbed?

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