XIII.- TRANSMISIÓN DE CALOR POR CONVECCIÓN CORRELACIONES PARA LA CONVECCIÓN NATURAL Y FORZADA http://libros.redsauce.net/
La complejidad de la mayoría de los casos en los que interviene la transferencia de calor por convección, hace imposible un análisis exacto, teniéndose que recurrir a correlaciones de datos experimentales; para una situación particular pueden existir diversas correlaciones procedentes de distintos grupos de investigación; además, con el paso del tiempo, determinadas correlaciones antiguas se pueden sustituir por otras más modernas y exactas, de forma que al final, los coeficientes de transferencia de calor calculados a partir de correlaciones distintas no son iguales, y pueden diferir, diferir, en general, general, en más de un 20% 20%,, aunque en circunstancias complicadas las discrepancias pueden ser mayores. En la convección natural, el fluido próximo a la pared se mueve bajo la influencia de fuerzas de empuje originadas por la acción conjunta de los cambios en su densidad y el campo gravitatorio terrestre.
XIII.1.- CORRELACIONES ANALÍTICAS PARA LA CONVECCIÓN NATURAL EN PLACA PLANA VERTICAL Uno de los problemas más simples y comunes de convección natural natural acontece cuando una superficie superficie vertical se somete a un enfriamiento o a un calentamiento mediante un fluido. Por comodidad supondremos que las capas límite térmica e hidrodinámica hidrodinámica coinciden coinciden Pr =1; =1; en principio, la capa límite es laminar, pero a una cierta distancia del borde, y dependiendo de las propiedades del fluido y del gradiente térmico, puede suceder la transición a régimen turbulento, lo cual sucede cuando
(Gr Pr) > 10 9, Fig XIII.1; el número de Grashoff es de la forma: Gr =
β Δ T L 3 g β Δ ν 2
;
β =
1 ( ∂v ) = 1 v - v F ; v ∂T p v F T - T F
1 Para un gas ideal: β = T º( º( K )
Dado que la convección natural es consecuencia de una variación de la densidad, el flujo correspondiente es un flujo compresible; pero, como la diferencia de temperaturas entre la pared y el fluido es pe-
u(x, y), u(x, y), v(x, y) y) como de la temperatura T(x, y) ρ g), en el que ρ debe T(x, y), considerando a la densidad constante, excepto en el término ( ρ queña, se puede hacer un análisis, tanto de las componentes de la velocidad
considerarse como función de la temperatura, ya que la variación de ρ en este término es el causante de XIII.-225
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la fuerza ascensional. La tercera ecuación de Navier-Stokes proporciona: proporciona:
1 dp = - g - du + ν Δu ρ dx dt ∂u + v ∂u ) = - ∂ p - ρ g + η ∂ 2 u ρ ( u ∂ x ∂ y ∂ x ∂ y 2 Gradiente de presiones a lo largo de la placa vertical:
∂ p = - ρ F g ; ∂ x
1 (- ρ g) = - g - du + ν Δ F ρ dt ν Δ u
siendo ρF la densidad del fluido fuera de la capa límite. Como el fluido al calentarse o enfriarse modifica su densidad, en el intervalo de temperaturas TF y T, se tiene:
g ( ρ F - ρ ) ) = ρ g (
Fig XIII.1.- Convección natural en placa vertical
ρ F ρ
- 1)
siendo ρF la densidad del fluido a la temperatura TF y ρ la densidad del fluido fluido del del interior interior de la capa límite a la temperatura T; como el volumen específico del fluido es:
v = v F { 1 + β (T - T F )} ; ρ ρ g ( F ρ
- 1)
= ρ g β ( T
ρ F = ρ
1 + β ( T - T F ) ⇒
ρ F ρ
-1
= β (T
- T F )
β Δ T - T F ) = ρ g β Δ
Teniendo en cuenta ecuaciones anteriores, la tercera ecuación de Navier-Stokes, (ecuación del momento), la ecuación de la energía y la ecuación de continuidad, quedan en la forma: 2 Ecuación del momento: u ∂ xu + v ∂ yu = g β ( T - T F ) + ν ∂ 2u ∂ ∂ ∂ y 2 Ecuación de la energía: energía: u ∂∂T + v ∂∂T = α ∂ 2T x y ∂ y u v Ecuación de continuidad: ∂∂ x + ∂∂ y = 0
Las condiciones de contorno para una placa vertical isoterma son:
y = 0 ; u = 0 ; v = 0 ; Para: y = ∞ ; u = 0 ; T = T ; F
T = T pF ∂u = 0 ; ∂T = 0 ∂ y ∂ y
Solución integral en pared isoterma.- La ecuación integral del momento de la cantidad de movimiento de la capa límite es:
∂ δ u 2 dy = δ g β (T - T ) dy + ν ∂ 2 u 〉 F ∂ x 0 ∂ y 2 y 0 0
∫
∫
=
en la que se ha supuesto que los espesores de las capas límite térmica e hidrodinámica son iguales.
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ficticia, función de x.
V = C1 x 4 δ = C 2 x
Las expresiones de V y δ se pueden poner en la forma:
Integrando las ecuaciones del momento y de la energía, resultan: 1 ∂ ( V 2 δ ) = 1 g β ( T δ - ν V ) T ) pF F 105 10 5 ∂ x 3 g T pF - T F ∂ ( V δ ) Ecuación de la energía: 2 α T ) ) = 1 ( T pF F δ ∂ x 30 Ecuación del momento:
V = C1 x 0,952 + Pr δ = 3,93 4 , resulta: 4 x Gr x Pr 2 δ = C 2 x
y teniendo en cuenta que:
ν V = 5 ,17 x
Gr x 0,952 + Pr
Q ∂T 〉 = h ( T - T ) = 2 k ( T - T ) ; h = 2 k = k pF F cF ∂ y y 0 cF pF F δ δ A =
Gr x Pr 2 4 Nu x Nu x = 0,508 4 0,952 ; Nu = ; Gr .Pr < 10 9 + Pr 3 Si, Ra > 109, el flujo comienza a ser turbulento, y suponiendo un perfil de velocidades (m = 7) se encuentra:
Nu x =
Gr x Pr 7/6 0,0295 ( ) 2/ 5 1 + 0,494 Pr 2/3
2/5 Nu = 0,021 Ra L
viniendo expresado hC en, Kcal/hora m2°C, la conductividad térmica k F del fluido en, Kcal/m°C y la velocidad másica G en, kg/m 2 hora.
Placa isotérmica.- Pohlhausen considera que los perfiles de velocidad y temperatura en convección natural presentan propiedades similares, en forma análoga a las observadas por Blasius para la convección forzada, de forma que: η
=
y 4 Gr x ; x 4
Φ =
T - T F y = ( 1 - δ ) 2 T pF - T F
La distribución de temperaturas permite determinar el flujo de calor local, de la forma:
Q A
=
k ( T - T ) 4 Gr x d Φ 〉 = h ( T - T ) - k ∂∂ yT 〉 y=0 = - x pF F 4 dη η=0 cF pF F
obteniéndose el número local de Nu
= f ( Pr) 4
Gr x
, viniendo los valores de f(Pr) en la Tabla XIII.1.
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Tabla XIII.1
Pr f(Pr)
0, 01 0,0 812
0, 72 0,5 046
0 ,73 3 0 ,50 8
1 0, 5671
2 0, 7165
10 1 ,1694
100 2,1 91
10 0 0 3, 966
Placa con flujo de calor constante.- Las ecuaciones del momento, energía y continuidad anterioQ res, son válidas para un flujo flujo de calor uniforme = Cte a lo largo de la placa; con esta condición se tie A ne:
Gr Nu = F ( P r ) 4 4 L , siendo : 0,95 F( Pr ) = 4 3 f ( P r ) Los valores de F(Pr) vienen dados en la Tabla XIII.2. Tabla XIII.2
Pr F(Pr)
0, 01 0, 335
1 0, 811
10 1, 656
100 3,0 83
XIII.2.- CORRELACIONES PARA LA CONVECCIÓN NATURAL EN PLACAS Para la determinación de los coeficientes de transmisión de calor por convección natural, con super-
ficie isoterma a Tp, en los casos de: - Pared vertical de altura L, (no se define la anchura) d > 35 - Tubo vertical con L 4 Gr L - Tubo horizontal de diámetro d se utiliza una ecuación general de la forma: Nu L =
Gr
=
g β ν 2
Δ T L 3 , y de Rayleigh: Ra =
C (Ra L ) n , que depende de los números de Grashoff:
Gr Pr
Las propiedades térmicas del fluido se toman a la temperatura media de la película, a excepción del coeficiente de dilatación térmica β que se evalúa a la temperatura del fluido TF. Para el caso de un gas ideal, el valor de β se puede aproximar por β ≅
1 T F (ºK)
ΔT es la diferencia entre la temperatura de la pared y la del fluido L es una longitud característica y los valores de C y
n vienen dados en las Tablas XIII.3.
Estas ecuaciones se pueden aplicar a la convección libre laminar desde placas verticales isotermas o superficies con flujo térmico uniforme, tomando la temperatura de la superficie superficie en el punto medio medio de la placa. Tablas XIII.3.- Valores de las constantes de la ecuación de Nusselt para convección natural Planos verticales y cilindros verticales
1700 Ra
106
1 08 Ra
1010
1 010 Ra
1013
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Superficie superior de placas calientes o superficie inferior de una placa fría
2.10 4 < Ra < 8.106 8.106 < Ra < 1011 C 0 ,5 4 0, 15 n 0 ,2 5 0, 33 Superficie inferior de placas calientes o superficie superior de placas frías
1 05 < Ra
<
1011
C = 0,58
n = 0,20
Para el estudio de la convección libre alrededor de placas planas rectangulares horizontales, se toma como longitud característica la media aritmética de sus dos dimensiones, o bien el 90% de su diámetro en el caso de discos circulares horizontales.
Convección natural sobre placa vertical.- El espesor de la capa límite viene dado por la expresión: δ
x
= 3,93 4
0,952 + Pr Gr x Pr 2
Gr x Pr 2 4 Nu x ; Gr Pr Pr < 10 9 Nu x = 0,508 4 0,952 + Pr ; Nu = 3 y el número de Nux por: Gr x Pr 7 /6 ) 2/5 ; Ra > 10 9 Nu x = 0,0295 ( 1 + 0, 494 Pr 2/ 3 2/5 ; Ra > 10 9 El nº de Nusselt Nusselt medio es: Nu L = 0,021 Ra L
Convección natural sobre placa vertical a temperatura uniforme.- Para determinar el coeficiente de convección natural en flujo laminar (Ra L < 108 ) con temperatura de pared vertical uniforme, se pueden utilizar los valores de la Tabla XIII.1: 8 n = 0,25 Nu L = C (Ra L ) n = C = 0,59 = 0,59 Ra L0,25 , para: 1700 < Ra L < 10 1 < Pr < 10
0,67 Ra L0,25 109 Ra L < 10 , para: o también: Nu L = 0,68 + 0,492 1 < Pr < 10 {1 + ( Pr ) 9/16 } 4/9 Para el flujo de transición laminar-turbulento (108 < Ra L < 1010 )
Nu L = C (Ra L ) n =
8 n = 0,33 1 010 = 0,13 Ra L0,33 , para: 10 < Ra L < 10 C = 0,13 1 < Pr < 10
Para flujos con turbulencia muy desarrollada (109 < Ra L < 1012 ) : n = 0,40 10 10 < Ra L < 10 1 013 Nu L = C (Ra L ) n = C = 0,021 = 0,021 Ra L0 ,4 , para: 1 < Pr < 10 Nu L = 0 68 +
0,67 Ra L0, 25
10 9 < Ra L < 10 12 {1 + 1,6.10 -8 Ra L ψ } }1/12 , para:
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to, hacia o desde una placa plana vertical de altura L, considerando en el eje de ordenadas NuL y en el eje de abscisas (Gr L Pr), que se pueden aplicar también al caso de cilindros verticales.
Fig XIII.2.- Capas límite laminar y turbulenta en la convección natural sobre paredes verticales
Fig XIII.3.- Correlación para la convección natural en placas y tubos verticales
Una expresión general que las engloba, válida tanto para régimen laminar como turbulento es: 0, 387 Ra L1/6 -1 Nu = 0,825 + 0,492 9/16 8/27 , para: 10 < Ra L {1 + ( Pr ) }
<
10 12
En la formulación propuesta, si una de las caras de la pared está aislada térmicamente, los valores del número de Nusselt serían la mitad de lo indicado en las fórmulas.
Para el caso particular del aire, a temperaturas normales, el coeficiente de transferencia de calor local para una placa vertical isotérmica se puede aproximar por las siguientes ecuaciones, teniendo en cuenta que para el aire la transición de régimen laminar a turbulento es Gr x 109: ≈
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El coeficiente de convección medio para toda la placa vertical es:
1 h c = L
L
1 { x cr t 1,07 4 Δ T dx + h c( x) dx = L x 0 0
∫
∫
í
L
∫ x
1, 3 3 ΔT dx}
cr ít
W m 2 º K K
Convección natural sobre placa vertical con flujo de calor uniforme.- En esta situación se utiliza un número de Grashoff modificado Gr x* , de la forma: Grr* =
g β q p x 4 Gr x Nu x = k ν 2
siendo q p el flujo de calor en la pared y: Nu x =
h cF x x k
Régimen laminar: laminar: Nu = 1,25 ( Nu x ) x=L ;
( )
Nu x = 0,60 (Gr x* Pr ) 1/5 ;
10 5 < Gr x* Pr < 10 11
Otra expresión para convección natural laminar, con flujo de calor uniforme es:
0 ,67 ( Gr L* Pr )1/4 5 * 11 0,492 9/16 4/9 ; 10 < Gr L Pr < 10 {1 + ( Pr ) } Régimen Régimen turbu turbulento lento:: Nu = 1 ,136 136 ( Nu x ) x=L ; Nu x = 0,568 (Gr x* Pr ) 0,22 ; 10 13 < Gr x* Pr < 1016
Nu Nu ( Nu - 0,68 0,68)) =
Convección natural sobre una placa inclinada un ángulo .- Si la placa caliente se inclina un pequeño ángulo θ respecto a la vertical, se puede tomar un número de Grashoff igual al número de Grashoff calculado para placa vertical multiplicado por cos θ, es decir:
Gr = Gr placa vertical cos θ θ < 88º Si la superficie caliente mira hacia arriba: Nu = 0 ,56 4 Gr L Pr cos θ , para: 10 5 < Ra < 10 11 L 1 0
Si la superficie caliente mira hacia abajo: 33 + 0 ,56 ( Gr Pr cos θ ) Nu = 0 ,145 145 ( Gr L Pr )0,33 - ( Gr c Pr )0 ,33 ) 0,25 c
θ = 15º ⇒ Gr c = 5.109 ; Gr L Pr < 10 11 ; Gr L > Gr c ; 108 ; θ = 60º ⇒ Gr c = 10
θ = 30º ⇒ θ = 75º ⇒
Gr c = 10 9 Gr c = 10 106
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Nu = 0,13 Ra L1/3 ; Ra L < 2.108 - Superficie caliente mirando hacia arriba: 1/3 8 11 , en las que L es Nu = 0,16 Ra L ; 5.10 < Ra L < 10 la longitud de los lados en el caso de placa cuadrada, o la longitud del lado más corto en el caso de placa rectangular. Cuando RaL= 107, se originan unas corrientes térmicas turbulentas irregulares sobre la placa dando como resultado un nº de Nu medio que no depende del tamaño ni de la forma de la placa
- Superficie caliente mirando hacia abajo: Nu = 0,58 Ra L0 ,2 ; 10 6 < Ra L < 10 11 , en la que las propiedades del fluido se toman a la temperatura:
T = T pF - 0,25 ( T pF - T F ) y las de β a la temperatura
media de película.
h El número de Nusselt medio es: Nu = cF k
L
=
q p L ( T pF - T F ) k
Existe una correlación general para placa horizontal que se calienta hacia abajo, con extensiones adiabáticas desarrollada por Hatfield y Edwards, como se muestra en la Fig XIV 5, de la forma:
10 6 < Ra < 10 10 13 A ) {(1 + X )0 ,39- X 0,39 } Ra0 ,13 Nu A = 6 ,5 (1 ( 1 + 0 ,38 L A , para: 0,7 < Pr < 4800 0 < a/A < 0,2
con:
Χ =
- 0,16 a )0 ,7 13,5 Ra A + 2,2 ( A
Fig XIII.4.- Convección natural laminar alrededor de una placa horizontal caliente
Fig XIII.5.- Esquema de una placa horizontal que se calienta hacia abajo en la que las extensiones adiabáticas están sombreadas
Convección natural entre placas horizontales.- Este caso se presenta cuando un fluido circula entre dos placas, como paredes con cámara de aire, o ventanas de doble vidrio, o paneles solares, etc. La
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Rad =
g β ( T h - T c ) d 3 ν α ν α
=
1708
y si la temperatura sigue aumentando, se van creando situaciones de flujo cada vez más complejas hasta que, finalmente, el flujo en el centro se vuelve turbulento. Fig XIII.6.- Convección natural celular en una capa horizontal de fluido confinado entre dos placas paralelas
Si se toma el aire como fluido, y considerando la placa inferior como la más caliente, Fig XIII.6, se tiene:
Nu = 0,195 Gr 0 ,25 , para: 10 4 < Gr < 4.10 5 Nu = 0,068 Gr 0 ,33 , para: 4.105 < Gr < 10 7 Tomando como fluido un líquido de número de Pr moderado, (como el agua), y considerando la placa inferior como la más caliente, se tiene: 407 , para: 3.105 < Ra < 7.10 9 Nud = 0,069 Gr d0,33 Pr 0 ,407 d
Convección natural entre placas verticales.- Para espacios confinados, en los que el fluido sometido a convección circula entre placas verticales de altura L, el efecto térmico se puede expresar como un simple cambio en la conductividad térmica del fluido. La circulación se da para cualquier valor de Rad > 0, y la transferencia de calor por conducción pura se efectúa para, Rad < 103. Al aumentar Ra d el flujo se desarrolla y se forman celdas de convección. Cuando Rad = 104 el flujo pasa a ser tipo capa límite, con capas que fluyen hacia arriba sobre la pared caliente y hacia abajo sobre la pared fría, mientras que en la región central el flujo permanece prácticamente estacionario. Cuando Rad = 105 se desarrollan hileras verticales de vórtices horizontales en el centro del flujo Cuando Rad = 106 el flujo en el centro se vuelve turbulento
Nu = 1 , para: Gr < 2.000 3 < Gr < 2.10 4 son: Nu L = 0,18 Gr 0,25 ( d )0,11 , para: 2.10 Valores de Nu para el aire con L > 3 , 2.1 0 d L Nu = 0,065 Gr 0,33 ( d )0 ,11 11 , para: para: 2.10 .1 0 4 < Gr < 107 L L
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1708 ( sen sen 1,8 1,8 θ ) )1 ,6 } + {( Rad cos θ )1/3 - 1 } Nu L = 1 + 1,44 ( 1 - Ra1708 ) { 1 Rad cos θ 5830 d cos θ en la que los términos entre corchetes deben hacerse cero si salen negativos.
b) θ = 60º ; 0 < Ra d < 107.- El valor de Nud se tomará el máximo de entre los siguientes: 0,175 d Nud = (0,104 + L ) Ra0,283 d
Nu7d =
0,0936 Rad0,314 7 1 + ( ) 0 ,5 1+ Rad 20 ,6 0 ,1 {1 + ( ) } 3160
6 0 Nu c) 60º < q < 90º ⇒ Nud = 90 - θ Nud(60º + θ - 60 d(60 º ) d(90º) 30 30
d) θ = 90º ; 10 3 < Rad < 107.- El valor de Nud se tomará el máximo de entre los siguientes: 293 0,104 Ra 0d ,293 Rad Nud = 3 1 + ( ) 3 6310 1 + ( ) 1,36 Rad Ra d Ra < 10 3 Nud = 0,242 ( Ld ) 0,272 , para: Nud 90º) = 1 d( 90º)
Nud = 0,0605 3
XIII.3.- CORRELACIONES PARA LA CONVECCIÓN NATURAL EN TUBOS
Convección natural sobre un tubo o un cilindro horizontal a) El número de Nusselt medio para la convección natural hacia y desde cilindros horizontales , se Nu = C ( Ra ) n en la que los valores de las constantes se pueden puede calcular a partir de la ecuación: Nu tomar de la Tabla correspondiente, o a partir de la gráfica de la Fig XIII.8. b) Unas expresiones más exactas son:
0,518 Ra 1d/4 10 −6 < Rad < 10 109 Para flujo laminar laminar:: Nud = 0,36 + , con: Pr > 0,5 9/16 } 4/9 {1 + ( 0,56 ) Pr Para flujo turbulento: turbulento: Nud = 0,60 + 0,387
Rad Rad > 10 9 , con: 6 0,56 Pr > 0 ,5 9/16 16/9 {1 + ( ) } Pr
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c) Para la transferencia de calor desde cilindros en posición horizontal hacia metales líquidos, se pue-
de utilizar: Nu = 0 ,53 4 Gr Pr 2 o también la ecuación de Baher: d) En convección
Nu = 0 ,445 4 Ra + 0,1183 8 Ra + 0 ,41 ; 10 -5 < Ra < 10 4
natural para el caso particular del aire y gases, para tubos horizontales y verticales
calientes, se puede aplicar la formulación : W Flujo laminar: h C = 1,18 4 ΔdT 2 m º K W 3 Flujo turbulento turbulento : h c = 1,65 ΔT 2
, con ΔT en ºC y d en metros m º K
Convección natural entre cilindros concéntricos.- El cilindro interior es el caliente y el cilindro exterior el frío; las correlaciones recomendadas para la convección natural se expresan en función de una conductividad térmica efectiva kefc que se sustituye en la ecuación de conducción correspondiente:
2 π k efec d ( T 1 - T 2 ) k efec 4 Pr Ra cil , con: = 0,386 ln ( r 2/r1 ) k 0,861 + Pr D 2 4 ( ln ) k D D 2 - D1 efec 1 10 2 < Ra cil < 10 7 ; > 1 ; Ra cil = 3 -3/5 Ra ; d = d k 2 d ( D1 + D 2-3/5 ) 5
Q=
XIII.4.- CORRELACIONES PARA LA CONVECCIÓN NATURAL EN ESFERAS
Esfera isoterma a) La transferencia de calor hacia y desde una esfera isoterma de diámetro d, en gases, viene dada por: Nud = 2 + 0 ,43 4 Ra d
1 011 1 < Ra d < 10 Pr ≈ 1
b) Para el caso particular de convección de una esfera isoterma en agua: Nud = 2 + 0 ,5 4 Ra d
3.105 < Ra d < 8.108 10 < Nu < 90
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k efec Pr Ra esf k efec 4 2 < Ra < 10 4 ; = 0,74 ; 10 esf k 0,861 + Pr k > 1 d Ra esf = D Rad 2 -7/5 -7/5 4 5 ( D ) ( D1 + D 2 ) 1 XIII.5.- CORRELACIONES PARA LA CONVECCIÓN FORZADA EN PLACAS
Flujo laminar sobre placa plana horizontal a) El número de Nusselt local en un flujo laminar sobre placa plana se verifica para valores del número de Re < 5.105 y viene dado por la ecuación de Pohlhausen:
Nu x = 0,332 Re x Pr 1/3 =
hC x x ; k
0 ,1 < Pr < 10 3
Una expresión exacta para el número de Nusselt medio, longitud L y flujo laminar es: 3 5 hC L Nu = k = 0 ,664 Re L Pr 1/3 , para: 10 < Re L < 5.10 Pr > 0,5
b) Una correlación exacta para metales líquidos es: Nu = 1 ,128 128 Re L Pr 1/3
Flujo laminar totalmente desarrollado desarrollado entre placas planas paralelas Coeficiente de rozamiento: λ =
96
; Re d < 2800 ; dh = 2 x separación entre entre placas
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Nu L = 0,036
{Re L0 ,8 -
9200}
η Pr 0,43 ( η F )0,25 pF
2.105 < Re L < 5,5.10 6 , para: 0 ,7 < Pr < 380 0,26 < ( η F /η pF ) < 3,5
siempre que la turbulencia sea pequeña. Las propiedades del fluido se evalúan a la temperatura media TF excepto ηpF que lo es a la temperatura de la pared. Para los gases las propiedades del fluido se evalúan a la temperatura de película. Si la turbulencia es elevada se puede eliminar el sumando 9200 obteniéndose resultados bastante razonables. c) Otra expresión del número de Nusselt medio para la longitud L viene dada por: 5 7 Re Nu L = 0 ,664 66 4 ReC Pr 1/3 + 0,036 Re L0 ,8 Pr 0,43 {1 - ( ReC ) 0 ,8 }, para: 5.10 < Re x < 3.10 0,7 < Pr < 400 L
El coeficiente de arrastre viene dado por la expresión:
C=
Re crít 1,328 Re crít 0,523 0,523 + ; Re crít < Re L < 10 1 09 2 2 Re Re ln (0,06 Re L ) Re crít L L ln (0,06 Re crít )
Capa limite turbulenta sobre una placa plana totalmente rugosa.- Se define un tamaño adimensional ε * del grano de arena en función de la rugosidad absoluta ε en la forma:
G ρ ε * =
ν
ε
C = ( 3 ,476 + 0 ,707 707 ln xε ) − 2,46 ; 150 < xε < 1,5.107 ; ε * > 60 x C x 2 , para: C L = ( 2 ,635 + 0 ,618 ln L ) − 2,57 ; 150 < L < 1,5.107 ; ε * > 60 ε ε
en la que G es el gasto másico y C x el coeficiente de arrastre. El criterio para determinar el tipo de régi-
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Para flujos completamente desarrollados en un tubo circular (L →
∞
) con temperatura de pared cons-
tante Nu = 3,656 Flujos no desarrollados.- El efecto de entrada del fluido en tuberías se manifiesta cuando las longitudes turbulentas iniciales sean mucho más cortas que en condiciones de régimen laminar o cuando el intercambio térmico comienza a efectuarse desde la entrada de la tubería y, por lo tanto, la capa límite térmica no está todavía desarrollada. a) Una ecuación que tiene en cuenta las longitudes térmica e hidrodinámica, Sieder y Tate, con tem-
peratura de pared constante es: η 14 , con : Gz = ( d Re Pr Nu = 1 ,86 3 Gz ( η F )0 ,14 Pr ) y Gz > 10 ; d L pF Pr > 0,5
siendo L la longitud del tubo y
3 Gz η > c
2
d el diámetro. Las propiedades del fluido que conducen al cálculo de Re y
Pr se calculan a la temperatura TF b) Otra expresión para el flujo a la entrada en un tubo circular en régimen laminar, con temperatura
de pared constante (Hausen): Nu = 3,66 +
0,0668 Gz 1 + 0,04 Gz 2/3
η c
y para el flujo a la entrada en un tubo circular en régimen laminar, con flujo de calor constante (Hausen):
Nu = 4,36 +
0,023 Gz 1 + 0,0012 Gz
η c
en la que las propiedades del fluido para calcular Re y Pr se toman a la temperatura TF.
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Tabla XIII.5.- Longitud de entrada térmica Lt e hidrodinámica Lh para flujo laminar por el interior de conductos de sección transversal circular y no circular
L t/dh Re Re Flujo térmico constante
L h/dh Re Re d
0,0 56
0,0 33
0 ,04 3
0,0 11
0,0 08
0 ,01 2
0,0 75 0,0 85 0 ,0 9
0,0 54 0,0 49 0,0 41
0 ,04 2 0 ,05 7 0 ,06 6
2b
2a 2b
a/b = 0,25 a/b = 0,50 a/b = 1,00
Flujo turbulento desarrollado por el interior de tuberías a) Los datos experimentales correspondientes a los estudios realizados sobre el movimiento en tubos de un gran número de líquidos, gases y vapores, se pueden expresar por las siguientes ecuaciones: En tubos lisos se aplica la ecuación de Dittus-Boelter:
0, 7 < Pr < 160 Nu = 0,023 Re 0,8 Pr a , para: L > 60, y Re > 10.000 d en la que se considera a = 0,4 para calentamientos y a = 0,3 para enfriamientos. b) Una correlación que permite una precisión aún mayor que la de Dittus-Boelter, es la de Polley, de la forma:
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que concuerda muy bien con los mejores datos experimentales para el aire y en un 10% con los mejores datos para números de Prandtl del orden de 103. e) En tubos rugosos se puede utilizar la analogía de Kàrmàn del capítulo anterior de la forma:
St = λ 8
1+ 5
λ
8
1 {( Pr - 1)
+ ln 5 Pr + 1 } 6
;
Pr < 30
f) En tubos rugosos también se puede utilizar la ecuación de Petukhov de la forma:
Red Pr λ η F n 1,07 + 12, 12 ,7 ( Pr 2/3 − 1 ) λ X 8 ( η pF ) ; X = 1,07 8 10 4 < Re < 5.10 6 ; 0 ,5 < Pr < 200 ; erro errorr < 5% 4 6 cuyo cuyo campo campo de validez validez es: es : 10 < Re < 5.10 ; 0 ,5 < Pr < 2000 ; erro errorr ≈ 10% 0 < η F /η pF < 40 Nud =
n = 0,11 para calentamiento con T pF uniforme n = 0,20 para enfriamiento con T pF uniforme n = 0 para flujo de calor uniforme o gases El valor del coeficiente de rozamiento viene dado para Pr > 0,5 por: λ = ( 0 ,79 ln λ =
Red - 1,64)-2 ; 10 4 < Red < 5.10 5.10 6
0,184 Re d−0 ,2 ; 2.10 4 < Re d < 3.10 5 , menos precisa que la anterior anterior
tomándose las propiedades del fluido a la temperatura media TF excepto ηpF que lo es a la temperatura de la pared TpF.
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λ =
1
e 5,02 R 13 - 2 lg { 7 R 4 4 - Re d lg ( 7 ,4 + Re d )} , ε
El criterio para determinar el tipo de régimen del flujo es: ( 0 < ε * < 5, 5 , liso ), ( 5 < ε * < 60, transi transici ci ón) y ( ε ε * > 60, 60, rug rugos osoo ) El número de Stanton local es: St
= λ 8
0,9 +
λ
8
1 { f ( e * , Pr) - 7,65}
en la que la función f( ε ε* , , Pr) depende de la rugosidad, presentando diversas formas, como se indica a continuación: ε * , Pr) = 4 ,8 Granos Granos de de arena: arena: f ( ε
ε * 0 ,2 Pr 0,44
ε * , Pr) = 4 ,8 Granos Granos de de arena: arena: f ( ε
ε * 0 ,28 Pr 0,57
Gene Genera rall : f ( ε * , Pr) = 0,55
ε *
; 1 < Pr < 6 ; 0 ,7 < Pr < 40
( Pr 2/3 - 1 ) + 9 ,5 ; Pr > 0 ,5 L h
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Flujo turbulento de metales líquidos por el interior de tuberías Flujo completamente desarrollado con flujo de calor uniforme desde la pared Nu = 0,625 Pe 0 ,4 , con: 10 2 < Pe Pe < 10 4 ; L > 60 d Nu = 4,82 + 0,0185 Pe 0,827 , con: 10 2 < Pe < 10 4 ; 3,6.10 3 < Re < 9.10 5 ; L 60 d > 60 Nud = 6 ,3 + 0,0167 Red0,85 Pr 0 ,93 , con: 10 4 < Red < 10 6 Flujo completamente desarrollado con temperatura de pared uniforme Red0,85 Pr 0 ,93 , con: 10 4 < Red < 10 6 Nu = 4,8 + 0,015 Pe 0,91 Pr 0 ,3 , con: con : Pr < 0 ,05 ; L d > 60 Nu = 5 + 0,05 Pe 0,77 Pr 0,25 , con: con : Pr Pr < 0, 0 ,1 ; Pe > 15.000 15.000 ; L > 60 60 d 08 , con: 0,004 < Pr < 0,1 ; Re < 500 Nu = 4,8 + 0,0156 Pe 0,85 Pr 0 ,08 50 0.000 ; L d Nud = 6 ,3
+ 0,0167
>
60
85 Pr 0,08 Para flujo de calor uniforme: Nu = 6 ,3 + 0 ,0167 Pe0 ,85 Flujo no desarrollado desarrollado : temperatura de pared pared uniforme uniforme:: Nu = 4 ,8 + 0 ,0156 Pe 0,85 Pr 0,08 Para temperatura
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considerar dos situaciones: a) Si se obliga al fluido a circular paralelo y pegado a la pared de las tuberías mediante pantallas, se considera como flujo por el exterior de tubos, y se utilizan para determinar el número de Nu las ecuaciones para un tubo único. b) Si no existen pantallas y los tubos están contenidos en una carcasa, se considera como flujo por el interior de un tubo, (la carcasa), introduciendo el concepto de diámetro equivalente en el número de Re de la formulación correspondiente que interviene en el cálculo del número de Nu. En esta situación, los números de Reynolds y Nusselt se calculan en función del diámetro hidráulico, en la forma:
Re =
u F d h ν
;
Nu =
Diámetro hidráulico:
hCF dh k F
dh = 4
Sección transersal mojada Perímetro mojado
Para una conducción formada por dos tubos concéntricos, Fig XIII.10.a:
d 2 - d 2 2 1 π 4
( d1 + d 2 ) ( d1 - d 2 )
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XIII.8.- CORRELACIONES PARA LA CONVECCIÓN FORZADA EN ESFERAS a) Para el flujo de fluidos sobre esferas con superficie isotérmica, se pueden utilizar los siguientes coeficientes de arrastre Cw referidos al diámetro d:
24 Cwd = Re
d
; Red < 0 ,5 2/3
24 ( 1 + Red ) ; 2 < Re < 500 Cwd = Re d 50 0 6 d Cwd = 0 ,44 ; 500 < Re d < 2.105 Whitaker propone una correlación general para el nº de Nusselt de la forma: Nud = 2 + ( 0,4 Re d + 0 ,06
3 Re 2 ) Pr 0 ,4 4 η F d η pF
3,5 ; 1<
< Red < 8.10 4 ; 0 ,7 < Pr < 380 η F η pF
< 3,2
calculándose las propiedades a la temperatura del fluido fluido TF excepto ηpF que se evalúa a la temperatura de la pared; para gases, el factor de corrección de la viscosidad es despreciable.
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Para convección combinada en tubos horizontales se pueden utilizar las siguientes expresiones:
Nu = 1,75
ηC
3
Gz +
0 ,0083 4 ( Gr
Pr ) 3
Re < 500 ; 10-2 < Pr d < 1 L , para: d Gz = Re Pr L
d )0 ,36 d <1 36 , para: Nu = 4,69 Re 0,27 Pr 0,21Gr 0,07 ( L para: Re > 500 ; 10-2 < Pr L Para la convección laminar combinada del agua que circula por un tubo horizontal con temperatura de pared constante, sus resultados vienen correlacionados a través de la expresión: η
Nud = 1,75 3 Gz + 0 ,012 01 2 3 ( Gz Gr d0,33 ) 4 ( η F )0,14 pF
Todas las propiedades del fluido se calculan a la temperatura media TF del fluido; esta ecuación da buenos resultados, siempre con un error menor del 8%. En la Fig XIII.3 se han representado los regímenes de convección libre, forzada y mixta en el caso de flujo por tubos horizontales.
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Tabla XIII.6.- Valores de n y C para tuberías cilíndricas en función del número de Re
Re (Para el diámetro d) 0,4 a 4 4 a 40 40 a 4.000 4.000 a 40.000 40.000 a 400 .000
C 0, 989 0, 911 0, 683 0, 193 0,02 66
n 0 ,33 0 0 ,38 5 0 ,46 6 0 ,61 8 0 ,80 5
Tabla XIII.7- Valores de n y C, función de la geometría del conducto
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Nud =
1 0,8 0,8237 - ln Red Pr
;
Re d Pr < 0,2
en la que las propiedades se evalúan a la temperatura del fluido TF.
Flujo cruzado en tubos en batería.- La transferencia de calor en la circulación de un fluido sobre una batería de tubos, en flujo cruzado, es muy importante por su aplicación al diseño y proyecto de la inmensa mayoría de los intercambiadores de calor. En la Fig XIII.12 se representan las líneas de corriente de un flujo laminar forzado alrededor de un cilindro, y en la Fig XIII.13, el flujo forzado a través de un haz
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de convección se multiplica por el factor de corrección ψ, que es igual a la unidad cuando el banco tubular está precedido por un codo, por una pantalla distribuidora o por un cortatiros. Tabla XIII.8.- Valores de C y n para baterías de 10 ó más tubos
EN LÍNEA ε y y/d 1 ,25 1, 5 2
ε x/d=
C 0 ,38 6 0 ,40 7 0 ,46 4
1,25 n 0 ,59 2 0 ,58 6 0 ,57 0
ε x/d=
C 0 ,30 3 0 ,27 8 0 ,33 2
1,5 n 0,6 08 0,6 20 0,6 02
ε x/d=
C 0,1 11 0,1 12 0,2 54
2 n 0,7 04 0,7 02 0,6 32
ε x/d=
C 0 ,070 3 0 ,075 3 0,2 20
3 n 0, 752 0, 744 0, 648
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Para líquidos, las propiedades se toman a TF excepto los números de Pr de la raíz, que lo son a las temperaturas respectivas. Para gases, las propiedades se toman a la temperatura de película; el término de la raíz que relaciona los números de Pr es aproximadamente la unidad. Para haces con menos de 20 tubos por fila, el número de Nud obtenido con la ecuación de Zukauskas se corrige mediante un factor de corrección Ψ que se determina a partir de la Fig XIII.15 en la forma:
Nu( N ) = Ψ Nu N >20
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En la Fig XIII.24 se presenta un ábaco que permite determinar el coeficiente de convección entre la
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XIII.11.- CORRELACIONES PARA LA CONVECCIÓN DE UN FLUJO A TRAVÉS DE UN LE-
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- El régimen es laminar desde r
hasta el eje de giro
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Convección forzada en el interior de tubos en régimen laminar.- Para la convección en tubos
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