S08 chap6 web
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This document provides an overview of convection heat transfer. It defines convection and the controlling equation for convection heat transfer. It describes factors that affect convection including geometry, flow type, boundary conditions, and fluid properties. It introduces important dimensionless groups used to analyze convection like the Prandtl number, Reynolds number, Grashof number, and Nusselt number. It then summarizes equations for forced convection over plates and cylinders/spheres. It also covers internal forced convection in circular tubes and natural convection over surfaces.
![N uL = 0.037
Re0.8
L
Pr
1/3
average, turbulent, UWT,
⇒ 0.6 < P r < 100, Rex > 500, 000
3. Combined Laminar and Turbulent Boundary Layer Flow, Isothermal (UWT)
N uL =
hL L
k
= (0.037 Re0.8 − 871) P r 1/3
L
average, combined, UWT,
0.6 < P r < 60,
⇒ 500, 000 ≤ ReL > 107
4. Laminar Boundary Layer Flow, Isoflux (UWF)
N ux = 0.453 Re1/2 P r 1/3
x
⇒ local, laminar, UWF, P r ≥ 0.6
5. Turbulent Boundary Layer Flow, Isoflux (UWF)
N ux = 0.0308 Re4/5 P r 1/3
x
⇒ local, turbulent, UWF, P r ≥ 0.6
Flow Over Cylinders and Spheres
1. Boundary Layer Flow Over Circular Cylinders, Isothermal (UWT)
The Churchill-Berstein (1977) correlation for the average Nusselt number for long (L/D > 100)
cylinders is
∗
N uD = SD + f (P r)
1/2
ReD
1
+
ReD
282, 000
5/8 4/5
average, UWT, Re < 107
⇒ 0 ≤ P r ≤ ∞, Re · P r > 0.2
∗
where SD is the diffusive term associated with ReD → 0 and is given as
∗
SD = 0.3
and the Prandtl number function is
f (P r) =
0.62 P r 1/3
[1 + (0.4/P r)2/3 ]1/4
6](https://image.slidesharecdn.com/s08chap6web-131127164217-phpapp01/85/S08-chap6-web-6-320.jpg)
![2. Isothermal Wall: The outlet temperature of the tube is
Tout = Tw − (Tw − Tin ) exp[−hA/(mCp )]
˙
Because of the exponential temperature decay within the tube, it is common to present the
mean temperature from inlet to outlet as a log mean temperature difference where
˙
Q = hA∆Tln
∆Tln =
ln
Tout − Tin
Tw − Tout
=
Tout − Tin
ln(∆Tout /∆Tin )
Tw − Tin
10](https://image.slidesharecdn.com/s08chap6web-131127164217-phpapp01/85/S08-chap6-web-10-320.jpg)
S08 chap6 web
- 1. Convection Heat Transfer Reading 12-1 → 12-8 13-1 → 13-6 14-1 → 14-4 Problems 12-41, 12-46, 12-53, 12-57, 12-76, 12-81 13-39, 13-47, 13-59 14-24, 14-29, 14-47, 14-60 Introduction • convection heat transfer is the transport mechanism made possible through the motion of fluid • the controlling equation for convection is Newton’s Law of Cooling ˙ Qconv = ∆T Rconv = hA(Tw − T∞ ) ⇒ where A = total convective area, m2 h = heat transfer coefficient, W/(m2 · K) 1 Rconv = 1 hA
- 2. Tw = surface temperature, ◦ C T∞ = fluid temperature, ◦ C Factors Affecting Convective Heat Transfer Geometry: flat plate, circular cylinder, sphere, spheroids plus many other shapes. In addition to the general shape, size, aspect ratio (thin or thick) and orientation (vertical or horizontal) play a significant role in convective heat transfer. Type of flow: forced, natural, mixed convection as well as laminar, turbulent and transitional flows. These flows can also be considered as developing, fully developed, steady or transient. Boundary condition: (i) isothermal wall (Tw = constant) or (ii) isoflux wall (qw = constant) ˙ Type of fluid: viscous oil, water, gases (air) or liquid metals. Fluid properties: symbols and units mass density specific heat capacity dynamic viscosity kinematic viscosity thermal conductivity thermal diffusivity Prandtl number volumetric compressibility : : : : : : : : ρ, (kg/m3 ) Cp , (J/kg · K) µ, (N · s/m2 ) ν, ≡ µ/ρ (m2 /s) k, (W/m · K) α, ≡ k/(ρ · Cp ) (m2 /s) P r, ≡ ν/α (−−) β, (1/K) All properties are temperature dependent and are usually determined at the film temperature, Tf = (Tw + T∞ )/2 External Flow: the flow engulfs the body with which it interacts thermally Internal Flow: the heat transfer surface surrounds and guides the convective stream Forced Convection: flow is induced by an external source such as a pump, compressor, fan, etc. 2
- 3. Natural Convection: flow is induced by natural means without the assistance of an external mechanism. The flow is initiated by a change in the density of fluids incurred as a result of heating. Mixed Convection: combined forced and natural convection Dimensionless Groups In the study and analysis of convection processes it is common practice to reduce the total number of functional variables by forming dimensionless groups consisting of relevant thermophysical properties, geometry, boundary and flow conditions. Prandtl number: P r = ν/α where 0 < P r < ∞ (P r → 0 for liquid metals and P r → ∞ for viscous oils). A measure of ratio between the diffusion of momentum to the diffusion of heat. Reynolds number: Re = ρU L/µ ≡ U L/ν (forced convection). A measure of the balance between the inertial forces and the viscous forces. Peclet number: P e = U L/α ≡ ReP r Grashof number: Gr = gβ(Tw − Tf )L3 /ν 2 (natural convection) Rayleigh number: Ra = gβ(Tw − Tf )L3 /(α · ν) ≡ GrP r Nusselt number: N u = hL/kf This can be considered as the dimensionless heat transfer coefficient. Stanton number: St = h/(U ρCp ) ≡ N u/(ReP r) Forced Convection The simplest forced convection configuration to consider is the flow of mass and heat near a flat plate as shown below. • as Reynolds number increases the flow has a tendency to become more chaotic resulting in disordered motion known as turbulent flow – transition from laminar to turbulent is called the critical Reynolds number, Recr Recr = U∞ xcr ν 3
- 4. – for flow over a flat plate Recr ≈ 500, 000 • the thin layer immediately adjacent to the wall where viscous effects dominate is known as the laminar sublayer Boundary Layers Velocity Boundary Layer • the region of fluid flow over the plate where viscous effects dominate is called the velocity or hydrodynamic boundary layer Thermal Boundary Layer • the thermal boundary layer is arbitrarily selected as the locus of points where T − Tw T∞ − Tw = 0.99 4
- 5. Flow Over Plates 1. Laminar Boundary Layer Flow, Isothermal (UWT) The local values of the skin friction and the Nusselt number are given as Cf,x = 0.664 Re1/2 x N ux = 0.332 Re1/2 P r 1/3 x N uL = hL L kf ⇒ local, laminar, UWT, P r ≥ 0.6 1/2 = 0.664 ReL Pr1/3 ⇒ average, laminar, UWT, P r ≥ 0.6 For low Prandtl numbers, i.e. liquid metals N ux = 0.565 Re1/2 P r 1/2 x ⇒ local, laminar, UWT, P r ≤ 0.6 2. Turbulent Boundary Layer Flow, Isothermal (UWT) Cf,x = τw 2 (1/2)ρU∞ N ux = 0.0296 = Re0.8 x 0.0592 Re0.2 x Pr 1/3 ⇒ local, turbulent, UWT, P r ≥ 0.6 local, turbulent, UWT, ⇒ 0.6 < P r < 100, Rex > 500, 000 5
- 6. N uL = 0.037 Re0.8 L Pr 1/3 average, turbulent, UWT, ⇒ 0.6 < P r < 100, Rex > 500, 000 3. Combined Laminar and Turbulent Boundary Layer Flow, Isothermal (UWT) N uL = hL L k = (0.037 Re0.8 − 871) P r 1/3 L average, combined, UWT, 0.6 < P r < 60, ⇒ 500, 000 ≤ ReL > 107 4. Laminar Boundary Layer Flow, Isoflux (UWF) N ux = 0.453 Re1/2 P r 1/3 x ⇒ local, laminar, UWF, P r ≥ 0.6 5. Turbulent Boundary Layer Flow, Isoflux (UWF) N ux = 0.0308 Re4/5 P r 1/3 x ⇒ local, turbulent, UWF, P r ≥ 0.6 Flow Over Cylinders and Spheres 1. Boundary Layer Flow Over Circular Cylinders, Isothermal (UWT) The Churchill-Berstein (1977) correlation for the average Nusselt number for long (L/D > 100) cylinders is ∗ N uD = SD + f (P r) 1/2 ReD 1 + ReD 282, 000 5/8 4/5 average, UWT, Re < 107 ⇒ 0 ≤ P r ≤ ∞, Re · P r > 0.2 ∗ where SD is the diffusive term associated with ReD → 0 and is given as ∗ SD = 0.3 and the Prandtl number function is f (P r) = 0.62 P r 1/3 [1 + (0.4/P r)2/3 ]1/4 6
- 7. All fluid properties are evaluated at Tf = (Tw + T∞ )/2. 2. Boundary Layer Flow Over Non-Circular Cylinders, Isothermal (UWT) The empirical formulations of Zhukauskas and Jakob given in Table 12-3 are commonly used, where N uD ≈ hD k = C Rem P r 1/3 D ⇒ see Table 12-3 for conditions 3. Boundary Layer Flow Over a Sphere, Isothermal (UWT) For flow over an isothermal sphere of diameter D N uD = ∗ SD + 0.4 1/2 ReD + 0.06 2/3 ReD Pr 0.4 µ∞ µw 1/4 average, UWT, 0.7 ≤ P r ≤ 380 ⇒ 3.5 < ReD < 80, 000 where the diffusive term at ReD → 0 is ∗ SD = 2 and the dynamic viscosity of the fluid in the bulk flow, µ∞ is based on T∞ and the dynamic viscosity of the fluid at the surface, µw , is based on Tw . All other properties are based on T∞ . 7
- 8. Internal Flow The Reynolds number is given as ReD = Um D ν For flow in a tube: ReD < 2300 laminar flow 2300 < ReD < 4000 transition to turbulent flow ReD > 4000 turbulent flow Hydrodynamic (Velocity) Boundary Layer • the hydrodynamic boundary layer thickness can be approximated as δ(x) ≈ 5x Um x ν −1/2 5x = √ Rex • the hydrodynamic entry length can be approximated as Lh ≈ 0.05ReD D (laminar flow) 8
- 9. Thermal Boundary Layer • the thermal entry length can be approximated as Lt ≈ 0.05ReD P rD (laminar flow) • for turbulent flow Lh ≈ Lt ≈ 10D Wall Boundary Conditions 1. Uniform Wall Heat Flux: Since the wall flux qw is uniform, the local mean temperature de˙ noted as Tm,x = Tm,i + qw A ˙ mCp ˙ will increase in a linear manner with respect to x. The surface temperature can be determined from Tw = Tm + qw ˙ h 9
- 10. 2. Isothermal Wall: The outlet temperature of the tube is Tout = Tw − (Tw − Tin ) exp[−hA/(mCp )] ˙ Because of the exponential temperature decay within the tube, it is common to present the mean temperature from inlet to outlet as a log mean temperature difference where ˙ Q = hA∆Tln ∆Tln = ln Tout − Tin Tw − Tout = Tout − Tin ln(∆Tout /∆Tin ) Tw − Tin 10
- 11. 1. Laminar Flow in Circular Tubes, Isothermal (UWT) and Isoflux (UWF) For laminar flow where ReD ≤ 2300 N uD = 3.66 ⇒ fully developed, laminar, UWT, L > Lt & Lh N uD = 4.36 ⇒ fully developed, laminar, UWF, L > Lt & Lh N uD = 1.86 ReD P rD L 1/3 µb µw 0.14 developing laminar flow, UWT, P r > 0.5 ⇒ L < Lh or L < Lt In all cases the fluid properties are evaluated at the mean fluid temperature given as Tmean = 1 2 (Tm,in + Tm,out ) except for µw which is evaluated at the wall temperature, Tw . 2. Turbulent Flow in Circular Tubes, Isothermal (UWT) and Isoflux (UWF) For turbulent flow where ReD ≥ 2300 the Dittus-Bouler equation (Eq. 13-68) can be used N uD = 0.023 Re0.8 P r n D turbulent flow, UWT or UWF, 0.7 ≤ P r ≤ 160 ReD > 2, 300 n = 0.4 heating ⇒ n = 0.3 cooling For non-circular tubes, again we can use the hydraulic diameter, Dh = 4Ac /P to determine both the Reynolds and the Nusselt numbers. In all cases the fluid properties are evaluated at the mean fluid temperature given as Tmean = 1 2 (Tm,in + Tm,out ) 11
- 12. Natural Convection What Drives Natural Convection? • fluid flow is driven by the effects of buoyancy • fluids tend to expand when heated and contract when cooled at constant pressure • therefore a fluid layer adjacent to a surface will become lighter if heated and heavier if cooled by the surface Recall from forced convection that the flow behavior is determined by the Reynolds number. In natural convection, we do not have a Reynolds number but we have an analogous dimensionless group called the Grashof number Gr = buouancy force viscous force = gβ(Tw − T∞ )L3 ν2 where g = gravitational acceleration, m/s2 12
- 13. β = volumetric expansion coefficient, β ≡ 1/T Tw = wall temperature, K T∞ = ambient temperature, K L = characteristic length, m ν = kinematic viscosity, m2 /s The volumetric expansion coefficient, β, is used to express the variation of density of the fluid with respect to temperature and is given as β=− 1 ∂ρ ρ ∂T P Natural Convection Over Surfaces • the velocity and temperature profiles within a boundary layer formed on a vertical plate in a stationary fluid looks as follows: 13
- 14. • note that unlike forced convection, the velocity at the edge of the boundary layer goes to zero Natural Convection Heat Transfer Correlations The general form of the Nusselt number for natural convection is as follows: N u = f (Gr, P r) ≡ CGr m P r n where Ra = Gr · P r 1. Laminar Flow Over a Vertical Plate, Isothermal (UWT) The general form of the Nusselt number is given as 1/4 N uL = hL kf =C gβ(Tw − T∞ )L3 ν2 ≡Gr 1/4 ν α 1/4 = C GrL P r 1/4 ≡P r Ra1/4 where gβ(Tw − T∞ )L3 RaL = GrL P r = αν 2. Laminar Flow Over a Long Horizontal Circular Cylinder, Isothermal (UWT) The general boundary layer correlation is 1/4 N uD = hD kf gβ(Tw =C − T∞ )D 3 ν2 ≡Gr 1/4 ν α ≡P r 1/4 = C GrD P r 1/4 1/4 RaD where RaD = GrD P r = gβ(Tw − T∞ )L3 αν All fluid properties are evaluated at the film temperature, Tf = (Tw + T∞ )/2. 14
- 15. Natural Convection From Plate Fin Heat Sinks Plate fin heat sinks are often used in natural convection to increase the heat transfer surface area and in turn reduce the boundary layer resistance R ↓= 1 hA ↑ For a given baseplate area, W × L, two factors must be considered in the selection of the number of fins • more fins results in added surface area and reduced boundary layer resistance, 1 R ↓= hA ↑ • more fins results in a decrease fin spacing, S and in turn a decrease in the heat transfer coefficient 1 R ↑= h↓A A basic optimization of the fin spacing can be obtained as follows: ˙ Q = hA(Tw − T∞ ) 15
- 16. where the fins are assumed to be isothermal and the surface area is 2nHL, with the area of the fin edges ignored. For isothermal fins with t < S Sopt = 2.714 S3L 1/4 RaS = 2.714 L 1/4 RaL with RaL = gβ(Tw − T∞ )L3 ν2 Pr The corresponding value of the heat transfer coefficient is h = 1.307k/Sopt All fluid properties are evaluated at the film temperature. 16