PROBLEMAS DE TRANSFERENCIA DE CALOR
Juan Carlos Ramos González Doctor Ingeniero Industrial Raúl Antón Remírez Doctor Ingeniero Industrial Diciembre de 2010
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Transferencia de Calor / Curso 2010-11
Problemas
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Transferencia de Calor / Curso 2010-11
Problemas
ÍNDICE Problemas Tema 1. Introducción a la transferencia de calor y a la conducción ............................. ............................. 1 Problemas Tema 2. Conducción unidimensional en régimen estacionario est acionario ..................................... ..................................... 7 Problemas Tema 3. Conducción bidimensional en régimen estacionario esta cionario ..................................... ..................................... 15 Problemas Tema 4. Conducción en régimen transitorio ............................................ ............................................................... ................... 21 Problemas Tema 5. Introducción a la convección................................................ ......................................................................... ......................... 29 Problemas Tema 6. Convección forzada forza da en flujo externo ................................................ ............................................................ ............ 33 Problemas Tema 7. Convección forzada en flujo interno ............................ ..................................................... ................................. ........ 37 Problemas Tema 8. Convección libre o natural ............................................ ................................................................... ................................ ......... 41 Problemas Tema 9. Introducción a la radiación ............................................. .................................................................... ............................... ........ 45 Problemas Tema 10. Intercambio Interca mbio radiativo entre superficies s uperficies .......................... ................................................ ............................. ....... 51
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Transferencia de Calor / Curso 2010-11
Problemas
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Transferencia de Calor / Curso 2010-11
Problemas
PROBLEMAS TEMA 1. INTRODUCCIÓN A LA TRANSFERENCIA DE CALOR Y A LA CONDUCCIÓN 1.
(2.7 del Incropera; Ley de Fourier) En el sistema mostrado en la figura se produce una conducción de régimen estacionario unidimensional sin generación de calor. La conductividad térmica es 25 W/m·K y el espesor L espesor L es es 0,5 m.
T 1
T 2
L
x
Determine las cantidades desconocidas para cada caso de la tabla siguiente y dibuje la distribución de temperatura indicando la dirección del flujo de calor. Caso
T 1
T 2
dT /dx (K/m)
q x′′ (W/m2)
1 400 K 300 K 2 100 ºC -250 3 80 ºC 200 4 -5 ºC 4.000 5 30 ºC -3.000 2 Solución: Solución: 1) 200 K/m, -5.000 W/m ; 2) 498 K, 6.250 W/m 2; 3) -20 ºC, -5.000 W/m 2; 4) -85 ºC, -160 K/m; 5) -30 ºC, 120 K/m. 2.
(1.13 del Incropera; Convección) Un chip cuadrado isotérmico de lado 5 mm está montado en un sustrato de manera que sus superficies laterales e inferior están bien aisladas, mientras que la superficie frontal se expone a la corriente de un fluido refrigerante a 15 ºC. La temperatura del chip no debe sobrepasar los 85 ºC. Si el fluido refrigerante es aire ( h = 200 W/m2·K), ¿cuál es la potencia máxima admisible del chip? Si el fluido refrigerante es un líquido dieléctrico (h (h = 3.000 W/m 2·K), ¿cuál es la potencia máxima admisible del chip? Solución: Solución: 0,35 W y 5,25 W.
3.
(Radiación y balance de energía) Un antiguo alumno de la Escuela que trabaja en la ESA
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Transferencia de Calor / Curso 2010-11
Problemas Stefan-Boltzmann σ = = 5,67·10 -8 W/m 2·K 4; temperatura de la superficie solar T s = 6.000 K; radio del Sol R s = 7·108 m; suponer que tanto el Sol como las placas se comportan como cuerpos negros (ε (ε = = α = = 1). Solución: Solución: Ro = R = R s (T (T s/T p)2. 4.
(Convección y radiación) Una persona desvestida tiene una superficie de 1,5 m2 expuesta a un ambiente y a unos alrededores de 27 ºC. La temperatura de su piel es de 33 ºC y se puede considerar un emisor de radiación perfecto. Si el coeficiente de transferencia de calor por convección es de 9 W/m 2·K, hállese: a) Las pérdidas de calor por convección y por radiación. b) El gasto energético en kcal/día. Solución: Solución: a) qconv = Q& conv = 81 W, qrad = Q& rad = 56,8 W; b) 2.846 kcal/día.
5.
(2.6 del Incropera; Ley de Fourier) Para determinar el efecto de la dependencia de la temperatura de la conductividad térmica sobre la distribución de temperatura en un sólido, considere un material para el que esta dependencia puede representarse como k = = k o + aT donde k o es una constante positiva y a es un coeficiente que puede ser positivo o negativo. Dibuje la distribución de temperatura de régimen estacionario asociada con la transferencia de calor en una pared plana para tres casos que corresponden a a > 0, a = 0 y a < 0.
6.
(2.11 del Incropera; Ley de Fourier) En el cuerpo bidimensional que se muestra en la figura se encuentra que el gradiente en la superficie A es ∂T /∂ y = y = 30 K/m. ¿Cuánto valen y y ∂T /∂ x en x en la superficie B? ∂T /∂ y y
Solución: Solución: ∂T /∂ y = y = 0; ∂T /∂ x = x = 60 K/m. 7.
(1.27 del Incropera; Balance de energía) Una placa de aluminio de 4 mm de espesor se monta en posición horizontal con su superficie inferior bien aislada. Se aplica a su superficie superior un recubrimiento que absorbe el 80% de cualquier radiación solar
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Transferencia de Calor / Curso 2010-11
Problemas de calor por convección entre la superficie y el aire es h = 20 W/m2·K. ¿Cuál es la velocidad inicial de cambio de la temperatura de la placa? Suponga que no hay alrededores. b) ¿Cuál será la temperatura de equilibrio de la placa cuando se alcancen las condiciones de régimen estacionario? c) Represente mediante Excel ® una gráfica de la temperatura de régimen estacionario como función de la emisividad para 0,05 ≤ ε ≤ 1, para tres valores de la absortividad de la placa de 0,5, 0,8 y 1 con el resto de condiciones constantes. Si la finalidad es maximizar la temperatura de la placa, ¿cuál es la combinación más deseable de emisividad y absortividad de la placa? Solución: Solución: a) 0,052 K/s; b) T s = 321 K. 8.
(1.31 del Incropera; Balance de energía) En una etapa de un proceso de recocido, 1 hoja de acero inoxidable AISI 304 se lleva de 300 K a 1.250 K conforme pasa a través de un horno calentado eléctricamente a una velocidad de v s = 10 mm/s. El espesor y ancho de la hoja son t s = 8 mm y w s = 2 m, respectivamente, mientras que la altura, ancho y largo del horno son H son H o = 2 m, W o = 2,4 m y Lo = 25 m, respectivamente. La parte superior y cuatro lados del horno se exponen al aire ambiental y a alrededores a 300 K, y la temperatura de la superficie del horno, su emisividad y el coeficiente de convección respectivos son T s = 350 K, ε s = 0,8 y h = 10 W/m2·K. La superficie inferior del horno también está a 350 K y reposa en una placa de cemento de 0,5 m de espesor cuya base está a 300 K. Estimar la potencia eléctrica que se requiere suministrar al horno. Datos: k cemento cemento (a 300 K) = 1,4 W/m·K. Propiedades termofísicas del acero inoxidable AISI 304: ρ = = 7.900 kg/m 3. T (K) (K) c p (J/kg·K) 600 557 800 582
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Transferencia de Calor / Curso 2010-11
Problemas T = = 100 – 150 x + x + 10 x2 donde T y x x tienen unidades de ºC y metros, respectivamente. Las variaciones de temperatura son insignificantes sobre la sección transversal de la columna. Evalúe la temperatura y la rapidez de conducción de calor en la unión columna-ducto ( x = x = 0) y en la interfaz columna-tierra ( x = x = 1). Explique la diferencia en las transferencias de calor. Solución: Solución: 18,75 W y 16,25 W. 10.
(2.17 del Incropera; Ley de Fourier) Un aparato para medir la conductividad térmica emplea un calentador eléctrico intercalado entre dos muestras idénticas de 30 mm de diámetro y 60 mm de longitud, prensadas entre placas que se mantienen a una temperatura uniforme T o = 77 ºC mediante la circulación de un fluido. Se pone grasa conductora entre todas las superficies para asegurar un buen contacto térmico. Se empotran termopares diferenciales en las muestras con un espaciado de 15 mm. Las caras laterales de las muestras se aíslan para que la transferencia de calor sea unidimensional.
a) Con dos muestras de acero inoxidable AISI 316 en el aparato, el calentador toma 0,353 A a 100 V y los termopares diferenciales indican ∆T 1 = ∆T 2 = 25,0 ºC. ¿Cuál es la conductividad térmica del material de la muestra de acero inoxidable y cuál la temperatura promedio de las muestras? Compare los resultados con los valores de la Tabla A.1 del Incropera.
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Transferencia de Calor / Curso 2010-11
Problemas Propiedades termofísicas del Armco: T (K) (K) k (W/m·K) (W/m·K) 300 72,7 400 65,7 Solución: Solución: a) k = = 15 W/m·K y T = 400 K; b) k = = 70 W/m·K y T = 380 K. 11.
(2.21 del Incropera; Ecuación de calor) En una varilla cilíndrica de 50 mm de diámetro de combustible de un reactor nuclear ocurre generación interna de calor a q&1 = 5·10 7 W/m3, y en condiciones de régimen estacionario la distribución de temperatura es T (r ) = a+br 2, donde T está está en grados Celsius y r en en metros, mientras a = 800 ºC y b = -4,167·10 5 ºC/m2. Las propiedades de la varilla de combustible son k = = 30 W/m·K, ρ = = 1.100 kg/m 3 y c p = 800 J/kg·K. a) ¿Cuál es la velocidad de transferencia de calor por unidad de longitud de la varilla en r = 0 (línea central) y en r = = 25 mm (superficie)? b) Si el nivel de potencia del reactor aumenta súbitamente a q& 2 = 10 8 W/m3, ¿cuál es la velocidad de cambio de temperatura en el tiempo inicial en r = = 0 y en r = = 25? Solución: Solución: a) q r ′ ( r = 0) = 0 y q r ′ (r = 25) = 9,8 × 10 4 W/m ; b) 56,8 K/s.
12.
(2.24 del Incropera; Ley de Fourier, ecuación de calor y balance de energía) Un estanque solar poco profundo con gradiente salino consiste en tres capas fluidas distintas y se utiliza para absorber energía solar. Las capas superior e inferior están bien mezcladas y sirven para mantener las superficies superior e inferior de la capa central a temperaturas uniformes T 1 y T 2, donde T 2 > T 1. Considere condiciones para las que la absorción de la radiación solar en la capa central proporciona una generación no uniforme de calor de la − forma q& ( x ) = Ae ax , y la distribución de temperatura en la capa central es: T ( x ) = −
A ka
2
e
− ax
+ Bx + C
Las cantidades A (W/m3), a (1/m), B (K/m) y C (K) son constante conocidas, y k es la conductividad térmica que también es constante.
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Transferencia de Calor / Curso 2010-11
Problemas
b) Determine si las condiciones son estacionarias o transitorias. c) Obtenga una expresión para la rapidez a la que se genera energía térmica en la capa central, por unidad de área superficial. Solución: a) q x′′ ( x = L ) = −
& ′′ = E gen
A a
(1 − e − ). aL
A a
e
− aL
− Bk ; q x′′ ( x = 0) = −
A a
− Bk ; b) Régimen estacionario; c)
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Transferencia de Calor / Curso 2010-11
Problemas
PROBLEMAS TEMA 2. CONDUCCIÓN UNIDIMENSIONAL EN RÉGIMEN ESTACIONARIO Pared plana 1.
(3.2 del Incropera) La ventana posterior de un automóvil se desempaña mediante el paso de aire caliente sobre su superficie interna. a) Calcular las temperaturas de las superficies interna y externa de una ventana de vidrio de 4 mm de espesor, siendo la temperatura del aire caliente T ∞,int = 40 ºC y su coeficiente de convección hint = 30 W/m 2·K y la temperatura del aire exterior T ∞,ext = 10 ºC y su coeficiente de convección hext = 65 W/m 2·K. b) Evalúe cualitativamente la influencia de T ∞,ext y hext sobre las temperaturas. Datos: k vidrio vidrio (a 300 K) = 1,4 W/m·K. Solución: Solución: a) T int int = 7,7 ºC y T ext ext = 4,9 ºC; b) Ambas disminuyen al aumentar hext y aumentan al aumentar T ∞,ext .
2.
(3.3 del Incropera) En la ventana posterior del automóvil del problema anterior se instala como sistema para desempañar su superficie interior un elemento de calentamiento consistente en una película transparente delgada con resistencias eléctricas. Al calentarse eléctricamente este dispositivo se establece un flujo de calor uniforme en la superficie interna. a) Calcular la potencia eléctrica por unidad de área de ventana necesaria para mantener la temperatura de la superficie interna a 15 ºC cuando la temperatura del aire interior es T ∞,int = 25 ºC y su coeficiente de convección hint = 10 W/m2·K. El aire exterior está en las mismas condiciones que en el problema anterior. b) Calcular la temperatura de la superficie externa de la ventana. c) Evalúe cualitativamente la influencia de T ∞,ext y hext sobre la potencia eléctrica. Solución: Solución: a) P ′′ = 1,27 kW/m 2; b) T = 11,1 ºC; c) P ′′ aumenta al aumentar h
y
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Transferencia de Calor / Curso 2010-11
Problemas Tablero de madera contraplacada: k (a (a 300 K) = 0,12 W/m·K.
a) Determine una expresión simbólica para la resistencia térmica total de la pared incluyendo los efectos de convección. b) Determine la pérdida de calor total de la pared. c) Si el viento soplara de manera violenta elevando hext a 300 W/m 2·K, ¿cuál sería el porcentaje de aumento relativo de la pérdida de calor? d) ¿Qué resistencia térmica influye en mayor medida sobre la pérdida de calor a través de la pared? Solución: Solución: b) 4.214 W; c) 0,45 %; d) La de la fibra de vidrio, que es el aislante y tiene la k
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Transferencia de Calor / Curso 2010-11
Problemas Pared cilíndrica 5.
(3.37 del Incropera) Un calentador eléctrico delgado se inserta entre una varilla circular larga y un tubo concéntrico con radios interior y exterior de 20 y 40 mm. La varilla A tiene una conductividad térmica k A = 0,15 W/m·K y el tubo B k B = 1,5 W/m·K. La superficie externa está en contacto con un fluido a temperatura T ∞ = -15 ºC y un coeficiente de convección de 50 W/m2·K. a) Determine la potencia eléctrica por unidad de longitud de los cilindros que se requieren para mantener la superficie externa del tubo B a 5 ºC. b) ¿Cuál es la temperatura en el centro de la varilla A?
Solución: a) 251 W/m; b) 23,4 ºC. 6.
(3.44 del Incropera) Una corriente eléctrica de 700 A fluye a través de un cable de acero inoxidable que tiene un diámetro de 5 mm y una resistencia eléctrica de 6·10 -4 Ω/m. El cable esté en un medio que tiene una temperatura de 30 ºC y el coeficiente total asociado con la convección y la radiación entre el cable y el medio es aproximadamente 25 W/m 2·K. a) Si el cable está expuesto, ¿cuál es la temperatura de la superficie? b) Si se aplica un recubrimiento muy delgado de aislante eléctrico al cable, con una resistencia de contacto de 0,02 m2·K/W, ¿cuáles son las temperaturas superficiales del aislante y del cable? c) Si se usa un aislante de conductividad térmica 0,5 W/m·K, ¿cuál será el espesor de este aislante que dará el valor más bajo de la temperatura del cable? ¿Cuál es el valor de esa temperatura?
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Transferencia de Calor / Curso 2010-11
Problemas Pared esférica 8.
(3.56 del Incropera) Una sonda esférica crioquirúrgica se incrusta en tejido enfermo con el propósito de congelarlo y destruirlo. La sonda tiene un diámetro de 3 mm y su superficie se mantiene a -30 ºC cuando se incrusta en tejido que está a 37 ºC. Se forma una capa esférica de tejido congelado alrededor de la sonda con una temperatura de 0 ºC en su superficie de contacto con el tejido normal. Si la conductividad térmica del tejido congelado es 1,5 W/m·K y el coeficiente de transferencia de calor por convección entre el tejido congelado y el normal es 50 W/m 2·K, ¿cuál es el espesor de la capa del tejido congelado? Resolución: Resolución: En primer lugar se expone una manera de obtener una expresión de la resistencia térmica para una esfera. En un elemento diferencial de esfera la aplicación de la conservación de la energía implica que qr = qr+dr , es decir que la transferencia de calor es independiente del radio, para condiciones unidimensionales de régimen estacionario y sin generación interna de calor. La ecuación de Fourier para una esfera hueca cuyas superficies están en contacto con fluidos a temperaturas distintas y en condiciones de régimen estacionario sin generación de calor adopta la forma: qr = − kA
dT dr
= −k (4π r 2 )
dT dr
donde A donde A = = 4πr 2 es el área normal a la dirección de la transferencia de calor. Al integrar la ecuación anterior: dr
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Transferencia de Calor / Curso 2010-11
Problemas La velocidad de transferencia de calor se puede expresar: q=
T ∞ − T s 2 1 4π h( r + e) 2
=
T s 2 − T s1 1 1 1 − 4π k r r + e
Al resolver se obtiene la siguiente ecuación en e2: e 2 + re −
k T s 2 − T s1 h T ∞ − T s 2
r = 0
Al resolver se obtiene: e = 5,34 mm. Generación interna de calor 9.
(3.73 del Incropera) El aire dentro de una cámara a T ∞,int = 50 ºC se calienta convectivamente con hint = 20 W/m2·K mediante una pared de 200 mm de espesor que tiene una conductividad térmica de 4 W/m·K y una generación de calor uniforme de 1.000 W/m3. Para prevenir que algo del calor generado se pierda hacia el exterior de la cámara, a T ∞,ext = 25 ºC con hext = 5 W/m2·K, se coloca un calentador de listón muy delgado sobre la pared exterior para proporcionar un flujo de calor uniforme, q o′′ .
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Transferencia de Calor / Curso 2010-11
Problemas externa del encamisado se expone a un fluido refrigerante que se caracteriza por una temperatura T ∞ y un coeficiente de convección h. a) Obtenga expresiones para las distribuciones de temperatura en el combustible y en el encamisado, T f (r ) y T c(r ). ). b) Considere un núcleo de combustible de óxido de uranio para el que K f = 2 W/m·K y r 1 = 6 mm y un encamisado para el que K c = 25 W/m·K y r 2 = 9 mm. Si q& = e& gen = 2·108 W/m3, h = 2.000 W/m2·K y T ∞ = 300 K, ¿cuál es la temperatura máxima en el elemento de combustible? c) Evalúe cualitativamente la influencia de h sobre las temperaturas. ¿Es posible mantener la temperatura de la línea central del combustible por debajo de 1.000 K ajustando el flujo de refrigerante y, por tanto, el valor de h? Solución: Solución: b) T f (r = = 0) = 1.458 K; c) Si h aumenta T f y T c disminuyen. No es posible. Superficies extendidas y aletas 11.
(3.109 del Incropera) Varillas de cobre circulares de diámetro D diámetro D = = 1 mm y longitud L longitud L = = 25 mm se usan para reforzar la transferencia de calor de una superficie que se mantiene a T s1 s1 = 100 ºC. Un extremo de la varilla se une a esta superficie (en x (en x = = 0) y el otro ( x ( x = = 25) se une a una segunda superficie que se mantiene T s2 s2 = 0 ºC. El aire que fluye entre las superficies también está a una temperatura T ∞ = 0 ºC y tiene un coeficiente de convección h = 100 W/m2·K. a) ¿Cuál es la transferencia de calor de una sola varilla de cobre? b) ¿Cuál es la transferencia total de calor de una sección de 1 m x 1 m de la superficie a
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Transferencia de Calor / Curso 2010-11
Problemas
Solución: Solución: q fo = 114,95 W; q fL = -88,08 W; qto = 5.972,5 W; qtL = -4.291,5 W. 13.
(3.131 del Incropera modificado, examen septiembre 2005) Se quiere disipar el calor
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Transferencia de Calor / Curso 2010-11
Problemas a) Dibuje el circuito térmico equivalente entre el interior del transformador y el aire exterior para la parte de pared que le corresponde a una aleta teniendo en cuenta que la dimensión perpendicular al dibujo es muy larga. b) Calcule los valores de las resistencias térmicas que aparecen en el circuito térmico anterior. c) Calcule la temperatura de la superficie interna del transformador, T int int . d) Calcule la temperatura de la superficie interna del soporte de aluminio (en contacto con la resistencia de contacto), T al al . e) Calcule la temperatura de la base de las aletas, T b. Solución: Solución: c) T int int = 532,3 K; d) T al al = 402,3 K; e) T b = 400,6 K.
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Transferencia de Calor / Curso 2010-11
Problemas
PROBLEMAS TEMA 3. CONDUCCIÓN BIDIMENSIONAL EN RÉGIMEN ESTACIONARIO Factores de forma 1.
(4.16 del Incropera) Con las relaciones de resistencia térmica desarrolladas en el tema 3 determine expresiones del factor de forma para las siguientes geometrías: a) Pared plana, capa cilíndrica y coraza esférica. b) Esfera hueca de superficie isotérmica de diámetro D diámetro D en en el interior de un medio infinito. Solución: Solución: a) A a) A// L, L, 2π L/ln( L/ln(r r 2/r 1), 4πr 1r 2/(r /(r 2-r 1); b) 2π D. D.
2.
(4.20 del Incropera) Un cable largo de transmisión de energía se entierra a una profundidad
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Transferencia de Calor / Curso 2010-11
Problemas
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Transferencia de Calor / Curso 2010-11
Problemas a) Calcule la temperatura de la unión y la transferencia de calor cuando el material de la base es (i) aluminio (k (k = = 240 W/m·K) y (ii) acero inoxidable (k ( k = = 15 W/m·K). b) Repita los cálculos anteriores para el caso del aluminio si entre la unión de la aleta y el − material de la base hay una resistencia térmica Rtc" = 3·10 5 m2·K/W c) ¿Cómo influye el coeficiente de convección en la transferencia de calor? Solución: Solución: a) (i) T j = 98 ºC, q f = Q& f = 4,44 W; (ii) T j = 78,4 ºC, q f = Q& f = 3,24 W; b) T j = 92 ºC, q f = Q& f = 4,08 W; c) Si h aumenta q f aumenta. 6.
(4.32 del Incropera) Se construye un iglú en forma de hemisferio con un radio interno de 1,8 m y paredes de nieve compactada de 0,5 m de espesor. En el interior del iglú el coeficiente de transferencia de calor por convección es 6 W/m2·K; en el exterior, en condiciones normales de viento, es 15 W/m 2·K. La conductividad térmica de la nieve compactada es 0,15 W/m·K. La temperatura de la capa de hielo sobre la que se asienta el iglú es de -20 ºC y tiene la misma conductividad térmica que la nieve compactada.
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Transferencia de Calor / Curso 2010-11
Problemas
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Transferencia de Calor / Curso 2010-11
Problemas
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Transferencia de Calor / Curso 2010-11
Problemas 10.
(4.52 del Incropera) Una barra larga de sección transversal rectangular tiene 60 mm de ancho, 90 mm de largo, y una conductividad térmica de 1 W/m·K. Uno de sus anchos está sometido a un proceso de convección con aire a 100 ºC y un coeficiente de convección de 100 W/m2·K. El resto de los lados se mantiene a 50 ºC.
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Transferencia de Calor / Curso 2010-11
Problemas
PROBLEMAS TEMA 4. CONDUCCIÓN EN RÉGIMEN TRANSITORIO Método de la resistencia interna despreciable
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Transferencia de Calor / Curso 2010-11
Problemas de la pared se cubre con una película delgada de cerámica que, para un área superficial unitaria, tiene una resistencia térmica de Rtf ′′ = 0,01 m 2·K/W. La superficie opuesta está bien aislada de los alrededores.
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