4. Free convection
Figure 1 Boundary layer development on a heated vertical plate. (a) Velocity and temperature profiles in
the boundary layer at the location x. (b) Boundary layer transitional flow conditions.
(a) laminar flow
(b) turbulent flow
Figure 2 Isotherms in natural convection over a hot plate in air.
The Volumetric Thermal Expansion Coefficient:
1⎛ ∂v ⎞
1 ⎛∂ρ ⎞
⎟⎟ = − ⎜⎜
⎟
β = ⎜⎜
ρ ⎝ ∂ T ⎟⎠ P
v ⎝ ∂ T ⎠P
For an ideal gas ( P = ρ R T ):
1 ⎛∂ρ ⎞
1 P
1
⎟⎟ =
=
β = − ⎜⎜
2
ρ ⎝ ∂ T ⎠P ρ RT
T
1
(a) A substance with a large β
(b) A substance with a small β
Figure 3 The coefficient of volume expansion is a measure of the change in volume of a substance with
temperature at constant pressure.
The Equation of Motion:
u
∂u
∂u
∂2 u
+v
= ν 2 + gβ (T − T∞ )
∂x
∂y
∂ y
The Grashof Number:
Gr =
g β (Ts − T∞ ) L3c
ν2
where
g = gravitational acceleration, m/s2
β = coefficient of volume expansion, 1/K (β = 1/T for ideal gases)
Ts = temperature of the surface, ◦C
T∞ = temperature of the fluid sufficiently far from the surface, ◦C
Lc = characteristic length of the geometry, m
υ = kinematic viscosity of the fluid, m2/s
Note:
The flow regime in forced convection is governed by the dimensionless Reynolds number, which
represents the ratio of inertial forces to viscous forces acting on the fluid. The flow regime in natural
convection is governed by the dimensionless Grashof number, which represents the ratio of the buoyancy
force to the viscous force acting on the fluid.
Figure 4 The Grashof number Gr is a measure of the relative magnitudes of the buoyancy force and the
opposing viscous force acting on the fluid.
2
The Rayleigh Number:
Ra = Gr Pr =
g β (Ts − T∞ ) L3c ν
α
ν2
=
g β (Ts − T∞ ) L3c
να
Note:
For vertical plates, the critical Rayleigh number is:
Ra x ,cr = Grx ,cr Pr ≈ 10 9
Correlations for Free Convections:
Nu L =
hL
= f (GrL , Pr) = f ( Ra)
k
Note:
- The values of the constants C and n depend on the geometry of the surface and the flow regime:
⎧1
for laminar flow
⎪
and usually C < 1 .
n = ⎨4
1
⎪
for turbulent flow
⎩3
- All properties are evaluated at the film temperature, T f = (Ts + T∞ ) 2
Boundary Layer Thickness:
For laminar flow of gases (Pr=0.7), the boundary layer thickness ( δ v ≈ δ T ) can be estimated using the
expression:
δ
= 6(Gr 4) −1 4
x
Pr = 0.7, Ra L ≤ 10 9
3
Table 1 Empirical correlations for the average Nusselt number for natural convection over surfaces.
4
Notes:
1. Inclined Plates:
(a) A hot inclined plate
(b) A cold inclined plate
Figure 5 Natural convection flows on the upper and lower surfaces of inclined plates.
2. Horizontal Plates:
As
p
where As and p are the plate surface area and perimeter, respectively.
Lc =
Figure 6 Free convection buoyancy-driven flows for hot (Ts > T∞) and cold (Ts < T∞) horizontal plates:
Case A — hot surface facing downwards, Case B — cold surface facing upwards, Case C — hot surface
facing upwards, and Case D — cold surface facing downwards.
3. Vertical Cylinders:
The relations for vertical plates can also be used for vertical cylinders if the boundary layer thickness is
much less than the cylinder diameter.
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4. Horizontal Cylinder:
Figure 7 Natural convection flow over a horizontal hot cylinder.
Combined Free and Forced Convection:
Figure 8 Variation of the local Nusselt number Nux for combined natural and forced convection from a
hot isothermal vertical plate.
Note:
For a given fluid, the parameter Gr/Re2 represents the importance of natural convection relative to forced
convection:
⎧ Gr Re 2 < 0.1
natural convection is negligible.
⎪
2
neither is negligible.
⎨0.1 < Gr Re < 10
⎪ Gr Re 2 > 10
forced convection is negligible.
⎩
6
(a) assisting flow
(b) opposing flow
(c) transverse flow
Figure 9 Natural convection can enhance or inhibit heat transfer, depending on the relative directions of
buoyancy-induced motion and the forced convection motion.
n
Nu combined = ( Nu nforced ± Nu natural
)1 n
+: for assisting and transverse flows
-: for opposing flow
3 < n < 4 and n = 3 for vertical sufaces.
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Example 1: Cooling of a Plate in Different Orientations
(a) Vertical
(b) Hot surface facing up
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(c) Hot surface facing down
9
Example 2: Horizontal cylinder: high pressure steam-line
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