TRANSFERENCIA DE CALOR POR CONVECCIÓN NATURAL

PROBLEMA

¿Cómo se conoce la tasa de transferencia de calor de  flujo de fluidos cuando esta ocurre por convección natural?

OBJETIVOS

Conocer los grupos de números adimensionales para calcular la transferencia de energía por convección natural.

Conocer los criterios para determinar la correlación que debe utilizarse

Usar hojas de Excel para realizar cálculos de transferencia de calor por convección natural  en tuberías y alrededor de objetos sumergidos.

Usar simuladores de Matemática para realizar cálculos de transferencia de calor por convección natural  en tuberías y alrededor de objetos sumergidos.

MENÚ

•Números adimensionales para la transferencia de energía por convección natural

•Cilindro horizontal

•Placas y cilindros verticales

•Esferas

•Correlaciones para otras geometrías

•Una nota de advertencia

•Cuestionario

PERFILES DE VELOCIDAD Y TEMPERATURA EN UNA PLACA VERTICAL

ECUACIONES PARA LOS PERFILES DE VELOCIDAD Y DE TEMPERATURA

Grid[{{Grid[{{Style["La ecuación de cantidad de movimiento",FontFamily->"Comic Sans MS",16]},{Style["ρ ( u ((∂u)/(∂x)) + v ((∂u)/(∂y)) ) = ( (Subscript[ρ, l]) ρ ) g + u (((∂^2)u)/(∂(y^2)))",FontFamily->"Comic Sans MS",16]},{Style["g ( (Subscript[ρ, l]) ρ ) = - g ρ β ( (Subscript[T, ∞]) - T )",FontFamily->"Comic Sans MS",16]},{Style["Donde es el coeficiente de dilatación térmica,definido como:",FontFamily->"Comic Sans MS",16]},{Style["β = - (1/ρ) ((∂ρ)/(∂T)) ≃ \!\(\*FractionBox[SubscriptBox[\(ρ\), \(∞\)], \(ρ\\ \((\\ T\\  - \\ \*SubscriptBox[\(T\), \(∞\)]\\ )\)\)]\)",FontFamily->"Comic Sans MS",16]},{Style["Para un gas ideal (es decir, ρ=(p/RT)), es el coeficiente de dilatación térmica es:",FontFamily->"Comic Sans MS",16]},{Style["β = - (1/(Subscript[T, ∞]))",FontFamily->"Comic Sans MS",16]}}]}}]

ECUACIONES PARA LOS PERFILES DE VELOCIDAD Y DE TEMPERATURA

perDeVelo=;
perDeTemp=;
perDeVelyTem=;
Grid[{{Grid[{{perDeVelo},{},{Style["u((∂u)/(∂x))+v((∂u)/(∂y))=gβ(T-(Subscript[T, ∞]))+v(((∂^2)u)/(∂(y^2)))",FontFamily->"Comic Sans MS",20]},{perDeTemp},{},{Style["u((∂T)/(∂x))+v((∂T)/(∂y))=α(((∂^2)u)/(∂(y^2)))",FontFamily->"Comic Sans MS",24]}}],perDeVelyTem}}]

s1=Framed[Manipulate[
Switch[ctrl,1,Grid[{
(*Prandatl*){Style["Pr",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(Cp*μ/k),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},
(*Reynols*){Style["Re",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(ρ*u*L/μ),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["δ",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(4.64*b/(((ρ*u*L/μ))^(1/2))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["δt",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2)))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["Para la velocidad",FontFamily->"Times New Roman",14]},{Style["a",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[0,FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["b",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[((3/2)*(u/(4.64*b/(((ρ*u*L/μ))^(1/2))))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["c",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[0,FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["d",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(-u/(2*(((4.64*b/(((ρ*u*L/μ))^(1/2))))^3))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},
{Style[Grid[{{"U(y)"}}],FontFamily->"TimesNewRoman",14],Style["=",FontFamily->"Times New Roman",14],Style[((3/2)*(u/(4.64*b/(((ρ*u*L/μ))^(1/2))))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]] , Style["y",FontFamily->"Times New Roman",14],Style[(-u/(2*(((4.64*b/(((ρ*u*L/μ))^(1/2))))^3))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]] , Style["(y^3)",FontFamily->"Times New Roman",14]},

{Style["Para la temperatura",FontFamily->"Times New Roman",14]},{Style["e",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(temps+273.15),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["f",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[((3/2)*((tempa+273.15)-(temps+273.15))/(0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2))))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["g",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[0,FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},{Style["h",FontFamily->"Times New Roman",14],Style["=",FontFamily->"Times New Roman",14],Style[(-((tempa+273.15)-(temps+273.15))/(2*(((0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2)))))^3))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]]},
{Style[Grid[{{"T(y)"}}],FontFamily->"TimesNewRoman",14],Style["=",FontFamily->"Times New Roman",14],Style[((temps+273.15)),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]],Style[((3/2)*((tempa+273.15)-(temps+273.15))/(0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2))))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]] , Style["y",FontFamily->"Times New Roman",14],Style[(-((tempa+273.15)-(temps+273.15))/(2*(((0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2)))))^3))),FontFamily->"Times New Roman",14,FontColor->RGBColor[0.68,0,0]] , Style["(y^3)",FontFamily->"Times New Roman",14]}


}],2,Plot[{((3/2)*(u/(4.64*b/(((ρ*u*L/μ))^(1/2)))))y+(-u/(2*(((4.64*b/(((ρ*u*L/μ))^(1/2))))^3)))(y^3),(temps+273.15)+((3/2)*((tempa+273.15)-(temps+273.15))/(0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2)))))y-(-((tempa+273.15)-(temps+273.15))/(2*(((0.976*(((Cp*μ/k))^(1/3))*(4.64*b/(((ρ*u*L/μ))^(1/2)))))^3)))(y^3)},{y,0,dx}]],TextCell[Text[Style["TRANSFERENCIA DE CALOR POR CONVECCIÓN NATURAL",Bold,18]]],{{ctrl,1," "},{1-> "Datos",2-> "Grafica" },ImageSize->Large},
{{dx,5,Style["Eje X",FontFamily->"TimesNewRoman",14]},0,100,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{L,5,Style["Longitud placa, m",FontFamily->"TimesNewRoman",14]},0,100,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{b,5,Style["Distancia del borde de ataque, m",FontFamily->"TimesNewRoman",14]},0,100,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},{{u,5,Style["Velocidad, m/s",FontFamily->"TimesNewRoman",14]},0,100,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{h,1,Style["Altura, m",FontFamily->"TimesNewRoman",14]},0,100,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{temps,30,Style["Temperatura de la Superficie, °C",FontFamily->"TimesNewRoman",14]},0,1000,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},{{tempa,30,Style["Temperatura del ambiente, °C",FontFamily->"TimesNewRoman",14]},0,1000,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{μ,1,Style["μ, kg (m^(-1)) (hr^(-1))",FontFamily->"TimesNewRoman",14]},0,10,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{ρ,1,Style["ρ, kg (m^(-3))",FontFamily->"TimesNewRoman",14]},0,10,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{Cp,1,Style["Cp, kcal (kg^(-1)) (K^(-1))",FontFamily->"TimesNewRoman",14]},0,10,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}},
{{k,1,Style["k, kcal (hr^(-1)) (m^(-1)) (K^(-1))",FontFamily->"TimesNewRoman",14]},0,10,Appearance->"Labeled",LabelStyle->{14,FontFamily->"TimesNewRoman"}}],Background->GrayLevel[0.85],RoundingRadius->8];

BÚSQUEDA DE LOS PARÁMETROS ADIMENSIONALES

inv=;
Grid[{{inv,Grid[{{Style["Las magnitudes físicas de interés son:",FontFamily->"Comic Sans MS",16]},{},{Grid[{

{Style["(Subscript[U, ∞])",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Velocidad característica",FontFamily->"Comic Sans MS",16]},

{Style["L",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Longitud",FontFamily->"Comic Sans MS",16]},

{Style["g",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Aceleración de gravedad",FontFamily->"Comic Sans MS",16]},

{Style["β",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Coeficiente de dilatación",FontFamily->"Comic Sans MS",16]},

{Style["(T-(Subscript[T, ∞]))",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Diferencia de temperatura",FontFamily->"Comic Sans MS",16]},

{Style["⋁",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Viscosidad cinemática",FontFamily->"Comic Sans MS",16]},

{Style["α",FontFamily->"Comic Sans MS",16],Style["=",FontFamily->"Comic Sans MS",16],Style["Difusividad térmica",FontFamily->"Comic Sans MS",16]}
},Frame->True,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]}}]}}]

DEPENDENCIA DEL NUSSELT

rey=;
pra=;
gra=;
Grid[{{Style["Utilizando el Teorema π: ",FontFamily->"Comic Sans MS",16]},{Style["Nu=Nu((Subscript[π, 1]),(Subscript[π, 2]),(Subscript[π, 3])",FontFamily->"Comic Sans MS",16]},{Grid[{{rey,Style["Reynols",FontFamily->"Comic Sans MS",16],Style["(Subscript[π, 1])=(((Subscript[U, ∞]) L)/ν)",FontFamily->"Comic Sans MS",16]},{pra,Style["Prandtl",FontFamily->"Comic Sans MS",16],Style["(Subscript[π, 2])=(ν/α)",FontFamily->"Comic Sans MS",16]},{gra,Style["Grashof",FontFamily->"Comic Sans MS",16],Style["(Subscript[π, 3])=((gβ (T - (Subscript[T, ∞])) (l^3))/(ν^2))",FontFamily->"Comic Sans MS",16]}}]}}]

DEPENDENCIA DE NU DE LOS NÚMEROS ADIMENSIONALES

Grid[{{Grid[{{Style["(Subscript[Gr, L])<<((Subscript[Re, L])^2)",FontFamily->"Comic Sans MS",16],Style["Convección forzada",FontFamily->"Comic Sans MS",16],Style["(Subscript[Nu, L])=f((Subscript[Re, L]),Pr)",FontFamily->"Comic Sans MS",16]},{Style["(Subscript[Gr, L])>>((Subscript[Re, L])^2)",FontFamily->"Comic Sans MS",16],Style["Convección libre",FontFamily->"Comic Sans MS",16],Style["(Subscript[Nu, L])=f((Subscript[Gr, L]),Pr)",FontFamily->"Comic Sans MS",16]},{Style["(Subscript[Gr, L])≈((Subscript[Re, L])^2)",FontFamily->"Comic Sans MS",16],Style["Convección combinada",FontFamily->"Comic Sans MS",16],Style["(Subscript[Nu, L])=f((Subscript[Re, L]),(Subscript[Gr, L]),Pr)",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]},{},{Style["En la transferencia de calor por convección, la viscosidad casi no interviene, por lo que el número de Brinkman no tendrá mucha relevancia.",FontFamily->"Comic Sans MS",16]}}]

DEPENDENCIA DE NU DE LOS NÚMEROS ADIMENSIONALES

Grid[{{Style["Nu= Ф (Gr)ψ(Pr)",FontFamily->"Comic Sans MS",16]},{},{Style["El número de Grashof y el número de Prandtl con frecuencia se agrupan como un producto GrPr, que se denomina número de Rayleigh, Ra. Después la relación del número de Nusselt se convierte en:",FontFamily->"Comic Sans MS",16]},{},{Style["Nu= Ф(Ra)",FontFamily->"Comic Sans MS",16]}}]

CONVECCIÓN LIBRE ALREDEDOR DE UN CILINDRO HORIZONTAL

graf=;
Grid[{{graf},{},{Style["En este caso Nu = Nu(Gr, Pr)",FontFamily->"Comic Sans MS",16]},{},{Style["Para Gr Pr > (10^4), esta gráfica está representada por la ecuación",FontFamily->"Comic Sans MS",16]},{},{Style["Nu  =  0.525 (GrPr)^(1/4)",FontFamily->"Comic Sans MS",16]}}]

EJEMPLO 1

cil=;
Grid[{{Style["Pérdida de calor por convección libre desde una tubería horizontal",FontFamily->"Comic Sans MS",16]},{},{Style["Estimar la velocidad de pérdida de calor por convección libre por unidad de longitud de una tubería horizontal de 15 cm de diámetro externo, si la temperatura de la superficie es de 38 C y el aire que la rodea está a 1atm y 27 C.",FontFamily->"Comic Sans MS",16]},{},{Style[cil,FontFamily->"Comic Sans MS",16]}}]

SOLUCIÓN

ingsol=;
tabla=;
tabla2=;
Grid[{{ingsol,Grid[{{Style["Primero calculamos la temperatura de película",FontFamily->"Comic Sans MS",16]},{},{Style["Tf=((Subscript[T, 0])+(Subscript[T, ∞]))/2=32.5=305.6 °K",FontFamily->"Comic Sans MS",16]},{},{Style["A una presión de 1 atm y esa temperatura, las propiedades del aire son:",FontFamily->"Comic Sans MS",16]},{tabla},{Style["Los demás valores que se necesitan son:",FontFamily->"Comic Sans MS",16]},{tabla2},{Style["Con estos valores calculamos el producto",FontFamily->"Comic Sans MS",16]},{},{Style["GrPr =(D^3) (ρ^2) g β ((Subscript[T, 0]) -(Subscript[T, ∞]))/ (µ^2)) (Cp µ/k)",FontFamily->"Comic Sans MS",16]},{Grid[{{Style["GrPr =((((((0.15))^3) (((1.158))^2) (1.27*(10^8)) (11/305.6))/0.0684))((0.241/0.0226))=3.2*(10^6)",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]},{Grid[{{Style["Usando la gráfica o la ecuación",FontFamily->"Comic Sans MS",16]},{Style["Nu = 0.525 (GrPr)1/4 se obtiene:",FontFamily->"Comic Sans MS",16]},{Style["Nu= 22.2 con lo cual",FontFamily->"Comic Sans MS",16]}}]},{Grid[{{Style["h=\!\(\*FractionBox[\(Nu\\ k\), \(D\)]\)=(((22.1) (0.0226))/0.15)=3.3 kcal (hr^(-1)) (m^(-2)) ((°C)^(-2))",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]}}]}}]

CORRELACIÓN PARA CONVECCIÓN NATURAL EN PLACAS Y CILINDROS VERTICALES

Ejemplo 2. Pérdida de calor por convección libre desde una Placa vertical

plac=;
Grid[{{Style["Un calentador que transfiere calor a una placa vertical de 15 centímetros de altura y 10 de ancho, trabaja rodeada de aire a 20 C.  ¿Qué cantidad de energía eléctrica debe suministrarse al calefactor para compensar las pérdidas por convección, si se quiere mantenerlo a 130 C?.",FontFamily->"Comic Sans MS",16]},{plac},{Grid[{{Style["(Nota: el análisis del problema completo requiere hacer el análisis también de la energía transmitida por radiación.)",FontColor->White,FontFamily->"Comic Sans MS",16]}},Background->RGBColor["#33cccc"]]}}]

SOLUCIÓN. PROPIEDADES FÍSICAS

tabla3=;
Grid[{{Style["En una tabla de propiedades del aire se obtienen los valores de μ,ρ,Cp,k y el Pr a la temperatura de película.Tf=(130+20)/2=75",FontFamily->"Comic Sans MS",16]},{tabla3}}]

Solución. Cálculo de Nu

Grid[{{Style["Con esos valores se calcula el número de Gr",FontFamily->"Comic Sans MS",16]},{},{Grid[{{Style["(Subscript[Gr, L]) = 65 (L^3) ((Subscript[T, s]) - (Subscript[T, ∞])) = 65 X (15 Cm)^3 X 110K  = 2.41 (X10^7)",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]},{},{Style["Para el aire a 75 C el Pr = 0.71 con lo que",FontFamily->"Comic Sans MS",16]},{Grid[{{Style["GrPr = 1.17 X (10^7)",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]},{},{Style["En ese caso podemos utilizar la relación:",FontFamily->"Comic Sans MS",16]},{Grid[{{Style["NuL = 0.555 ((Subscript[Gr, L]) Pr)^(1/4)",FontFamily->"Comic Sans MS",16]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]]]}}]

Solución. Cálculo de q

Grid[{{Style["NuL=Que sustituyendo los valores numéricos da: 35.7",FontFamily->"Comic Sans MS",16]},{},{Style["Despejando",FontFamily->"Comic Sans MS",16]},{},{Style["h=35.7(k/L)=((2.9*(10^(-2)))/0.15)=6.9 (W/((m^2) K))",FontFamily->"Comic Sans MS",16]},{},{Style["Con lo que:",FontFamily->"Comic Sans MS",16]},{},{Style["q = A (Ts -T∞) = (2 X 0.15 X 0.10) X 6.9 X 110 = 22.77 W
",FontFamily->"Comic Sans MS",16]}}]

ESFERA SUMERGIDA. CONVECCIÓN LIBRE

CORRELACIONES PARA OTRAS SITUACIONES

plac=;plachor=;plachorab=;cillarhorz=;cilinincl=;conover=;placasvert=;placashor=;esfera=;esferasconce=;cilrota=;disrota=;
esferota=;
Grid[{{Style["Geometría",FontColor->White,FontFamily->"Comic Sans MS",16],Style["Ecuación de Correlación",FontColor->White,FontFamily->"Comic Sans MS",16],Style["Restricciones",FontColor->White,FontFamily->"Comic Sans MS",16]},{Grid[{{Style["Placa larga vertical o inclinada con la superficie caliente hacia abajo.",FontFamily->"Comic Sans MS",12]},{plac}}],Style["NuL= 0.56(GrL Pr cos θ )^(1/4)",FontFamily->"Comic Sans MS",12],Grid[{{Style["(10^5)<(Subscript[Gr, L])Pr cos θ<(10^11)",FontFamily->"Comic Sans MS",12]},{Style["0 ≤ θ ≤ 89°",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Placa larga horizontal con la superficie caliente hacia arriba o la superficie fría hacia abajo",FontFamily->"Comic Sans MS",12]},{plachor}}],Grid[{{Style["NuL= 0.54 ((Subscript[Ra, L])^(1/4))",FontFamily->"Comic Sans MS",12]},{},{Style["NuL= 0.15 ((Subscript[Ra, L])^(1/3))",FontFamily->"Comic Sans MS",12]},{},{Style["L=A/P",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["(10^5)< (Subscript[Ra, L]) <(10^7)",FontFamily->"Comic Sans MS",12]},{},{Style["(10^7)< (Subscript[Ra, L]) <(10^10)",FontFamily->"Comic Sans MS",12]},{},{}}]},{Grid[{{Style["Placa horizontal con la superficie caliente hacia abajo o la superficie fría hacia arriba",FontFamily->"Comic Sans MS",12]},{plachorab}}],Grid[{{Style["(Subscript[Nu, L])= 0.27 ((Subscript[Ra, L])^(1/4))",FontFamily->"Comic Sans MS",12]},{},{Style["L=A/P",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["(10^5)< (Subscript[Ra, L]) <(10^7)",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Un cilindro horizontal largo",FontFamily->"Comic Sans MS",12]},{cillarhorz}}],Grid[{{Style["(Subscript[Nu, D])= 0.53((Subscript[Gr, D]) Pr)^(1/4)",FontFamily->"Comic Sans MS",12]},{},{Style["(Subscript[Nu, D])= 0.53((Subscript[Gr, D]) (Pr^2))^(1/4)",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["Pr > 0.5(X10^3)   GrD < (10^9)",FontFamily->"Comic Sans MS",12]},{},{Style["Metales líquidos, flujo laminar",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Cilindro inclinado, longitud L",FontFamily->"Comic Sans MS",12]},{cilinincl}}],Grid[{{Style["(Subscript[Nu, L]) = [ 2.9 − 2.32(sen θ)^0.8]",FontFamily->"Comic Sans MS",12]},{Style["x ((Subscript[Gr, D])) − 1/12[(Subscript[Gr, L])pPr]^((1/4 + 1/12 (senθ) 1.2))",FontFamily->"Comic Sans MS",12]},{},{Style["NuL=[0.47+0.11(senθ)^0.8]((Subscript[Gr, D]))^(-1/12)((Subscript[Gr, L])Pr)^(1/3)",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["Laminar:",FontFamily->"Comic Sans MS",12]},{Style["9.88 ×(10^7) ≤ (Subscript[Gr, L])Pr ≤ ((Subscript[Gr, L])Pr)cr",FontFamily->"Comic Sans MS",12]},{Style["1.08 ×(10^4)≤ (Subscript[Gr, L])Pr ≤6.9 ×\!(10^5)",FontFamily->"Comic Sans MS",12]},{},{Style["Turbulento:",FontFamily->"Comic Sans MS",12]},{Style["((Subscript[Gr, L])Pr)cr ≤ (Subscript[Gr, L])Pr ≤ 2.95×(10^10)",FontFamily->"Comic Sans MS",12]},{Style["1.08 ×(10^4)≤ (Subscript[Gr, D]) ≤ 6.9 ×(10^5)",FontFamily->"Comic Sans MS",12]},{Style["Donde ((Subscript[Gr, L]) Pr)_cr = 2.6×(10^9) + 1×(10^9) tanθ",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Cono vertical",FontFamily->"Comic Sans MS",12]},{conover}}],Grid[{{Style["Nu = 0.63 ( 1 + (Subscript[0.72, E]))((Subscript[Gr, L])^(1/4))",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["3° < Ø < 12°",FontFamily->"Comic Sans MS",12]},{Style["7.5 < log (Subscript[Gr, L]) < 8.7",FontFamily->"Comic Sans MS",12]},{Style["0.2 ≤ ϵ < 0.8",FontFamily->"Comic Sans MS",12]},{Style["Donde",FontFamily->"Comic Sans MS",12]},{Style["ϵ = 2/[((Subscript[Gr, L])^(1/4)) tan(Ø/z)]",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Espacio contenido entre dos placas verticales contenido desde un lado.",FontFamily->"Comic Sans MS",12]},{placasvert}}],Grid[{{Style["(Subscript[Nu, δ])=0.22((L/δ))((Pr/(0.2 + Pr))(Subscript[Ra, δ]))^0.28",FontFamily->"Comic Sans MS",12]},{},{Style["(Subscript[Nu, δ])=0.18((Pr/(0.2 + Pr))(Subscript[Ra, δ]))^0.29",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["2 < (L/8) < 10, Pr < 10",FontFamily->"Comic Sans MS",12]},{Style["(Subscript[Ra, δ]) < (10^10)",FontFamily->"Comic Sans MS",12]},{},{Style["1 ≤ (L/8) < 2, (10^(-3)) < Pr <(10^5)",FontFamily->"Comic Sans MS",12]},{Style["(10^3) < (RaPr/(0.2 + Pr))",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Espacio contenido entre dos placas horizontales calentado desde abajo",FontFamily->"Comic Sans MS",12]},{placashor}}],Grid[{{Style["Nu = 1 + 1.44 [ 1 − (1708/(Subscript[Ra, δ])) + [((Subscript[Ra, δ])/5830)]^(1/3) - 1 ]",FontFamily->"Comic Sans MS",12]},{},{Style["Nu = 1 + 1.44 [ 1 − (1708/(Subscript[Ra, δ])) + [((Subscript[Ra, δ])/5830)]^(1/3) - 1 ] + 20(((Subscript[Ra, δ])/140))(1-ln(((Subscript[Ra, δ])^(1/3))/140))",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["Aire",FontFamily->"Comic Sans MS",12]},{Style["1700 < (Subscript[Ra, δ]) < (10^8)",FontFamily->"Comic Sans MS",12]},{},{Style["Agua",FontFamily->"Comic Sans MS",12]},{Style["1700 < (Subscript[Ra, δ]) <3.5(x10^9)",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Interior de una cavidad esférica",FontFamily->"Comic Sans MS",12]},{esfera}}],Grid[{{Style["(Subscript[Nu, D]) = C ((Subscript[Gr, D]) Pr)^n",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["GrD",FontFamily->"Comic Sans MS",12],Style["c",FontFamily->"Comic Sans MS",12],Style["n",FontFamily->"Comic Sans MS",12]},{Style["(10^4) - (10^9)",FontFamily->"Comic Sans MS",12],Style["0.59",FontFamily->"Comic Sans MS",12],Style["1/4",FontFamily->"Comic Sans MS",12]},{Style["(10^9) - (10^12)",FontFamily->"Comic Sans MS",12],Style["0.13",FontFamily->"Comic Sans MS",12],Style["1/3",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Esferas concéntricas",FontFamily->"Comic Sans MS",12]},{esferasconce}}],Grid[{{Style["(Subscript[K, ef])=0.74 (((b^(1/4))/((Subscript[D, 0]) (Subscript[D, i]) (((((Subscript[D, i])^(7/5)) + (Subscript[D, 0])))^(5/4))))) ((Subscript[xRa, b])^(1/4))((Pr/(0.861 + Pr)))^(1/4)",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["0.70 < Pr <4200",FontFamily->"Comic Sans MS",12]},{Style["10 < ((b/((Subscript[D, 0]) (Subscript[D, i]) ((((Subscript[D, i]) + (Subscript[D, 0])))^5))))(Subscript[Ra, b]) < (10^7)",FontFamily->"Comic Sans MS",12]}}]},(*******************){Grid[{{Style["Cilindro rotatorio largo",FontFamily->"Comic Sans MS",12]},{cilrota}}],Grid[{{Style["(Subscript[Nu, D])=(((Subscript[h, c]) D)/k)=0.11(0.5((Subscript[Re, w])^2)+(Subscript[Gr, D])Pr)^0.35",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["(Subscript[Re, w])=((((πD)^2) ω)/v)>8000",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Disco rotatorio",FontFamily->"Comic Sans MS",12]},{disrota}}],Grid[{{Style["(Subscript[Nu, D])=(((Subscript[h, c]) D)/k)=0.36((Subscript[Re, w]))^(1/2)",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["(Subscript[Re, w])=\!\(\*FractionBox[\(ω\\ \*SuperscriptBox[\(D\), \(2\)]\), \(v\)]\)<(10^6)",FontFamily->"Comic Sans MS",12]}}]},{Grid[{{Style["Disco rotatorio",FontFamily->"Comic Sans MS",12]},{esferota}}],Grid[{{Style["(Subscript[Nu, D])=0.43((Subscript[Re, w])^0.5)(Pr^0.4)",FontFamily->"Comic Sans MS",12]},{},{Style["(Subscript[Nu, D])=0.066((Subscript[Re, w])^0.67)(Pr^0.4)",FontFamily->"Comic Sans MS",12]}}],Grid[{{Style["(Subscript[Re, w])=\!\(\*FractionBox[\(ω\\ \*SuperscriptBox[\(D\), \(2\)]\), \(v\)]\)<5(x10^4)",FontFamily->"Comic Sans MS",12]},{},{Style["Pr>0.7",FontFamily->"Comic Sans MS",12]},{},{Style["5(x10^4)<(Subscript[Re, ω])<7(x10^5)",FontFamily->"Comic Sans MS",12]}}]}},Frame->All,FrameStyle->Directive[Thickness[2],RGBColor["#33cccc"]],Background->{None,{RGBColor["#33cccc"],None}}]

CUIDADO CON LAS CORRELACIONES

Style["5(x10^4)<(Subscript[Re, ω])<7(x10^5)",FontFamily->"Comic Sans MS",12]

cuicor=;
Grid[{{cuicor},{Button[Mouseover[Style["http://tylervigen.com/spurious-correlations","Hyperlink"],Style["http://tylervigen.com/spurious-correlations","HyperlinkActive"]],NotebookLocate[{URL["http://tylervigen.com/spurious-correlations"],None}],Appearance->None]}}]

Trabajo realizado con el apoyo del Programa UNAM-DGAPA-PAPIME PE110517