SME
Heat Transfer · Problem type
Free convection on a vertical plate
Use this for “A 0.5 m tall plate is 40 K warmer than still air; film 320 K, ν = 1.75 × 10⁻⁵, α = 2.5 × 10⁻⁵ m²/s, k = 0.028. Find h.”
Diagram
Ra
=
gΔT
L
3
Tf να
Ra
=
9.81
×
40
×
0.5
3
320 × 1.75 ×
10
-5
× 2.5 ×
10
-5
Ra
=
3.504 ×
10
8
Nu = 0.59Ra1/4 = 80.72
h
=
Nu
k
L
=
4.52
W/(
m
2
·K)
Warmed air by the plate rises in a layer that thickens as x1/4 up the plate.
This is an example. Type your own values to start a new problem.
Ra
=
9.81
×
ΔT (K)
×
(
L (m)
)
3
÷
Tf (K)
÷
ν (m²/s)
÷
α (m²/s)
=
0.59
×
Ra
0.25
h (W/(m²·K))
=
Nu
×
k (W/(m·K))
÷
L
Formulas
Ra
=
9.81
×
ΔT
×
L
3
Tf ×
ν
×
α
3.5036 ×
10
8
=
9.81
×
40
×
0.5
3
320 × (1.75 ×
10
-5
) × (2.5 ×
10
-5
)
Nu = 0.59 × Ra0.25
80.7197
=
0.59
×
(3.5036 ×
10
8
)
0.25
h
=
Nu
×
k
L
4.5203
=
80.7197
×
0.028
0.5
Assumptions
•
Air as an ideal gas: β = 1 ÷ Tf, with Tf the film temperature in kelvins.
•
Laminar free convection: Ra between 10⁴ and 10⁹; above 10⁹ it is turbulent (Churchill–Chu).
•
ν, α and k are read at Tf; g = 9.81 m/s².
Step-by-step
WE KNOW
ΔT = 40 K, L = 0.5 m, Tf = 320 K, ν = 1.75 × 10⁻⁵ m²/s, α = 2.5 × 10⁻⁵ m²/s, k = 0.028 W/(m·K)
FIND
Rayleigh number (Ra), average Nusselt number (Nu), average film coefficient (h)
Step 1 · Find Rayleigh number (Ra)
Use Ra = 9.81 × ΔT × L³ ÷ (Tf × ν × α)
Buoyancy
gβΔT
L
3
over
the
drag
of
viscosity
and
diffusion;
for
a
gas
β
is
1
Tf
.
Ra
=
9.81
×
40
×
0.5
3
320 × (1.75 ×
10
-5
) × (2.5 ×
10
-5
)
Ra
=
9.81
×
40
×
0.5
3
0.0056 × (2.5 ×
10
-5
)
Ra
=
9.81
×
40
×
0.5
3
1.4 ×
10
-7
Ra
=
9.81
×
40
×
0.125
1.4 ×
10
-7
Ra
=
392.4
×
0.125
1.4 ×
10
-7
Ra
=
49.05
1.4 ×
10
-7
Ra
=
3.50357 ×
10
8
Step 2 · Find average Nusselt number (Nu)
Use Nu = 0.59 × Ra0.25
In laminar free convection Nu grows as the fourth root of Ra.
Nu
=
0.59
×
(3.50357 ×
10
8
)
0.25
Nu = 0.59 × 136.8
Nu = 80.7197
Step 3 · Find average film coefficient (h)
Use h = Nu × k ÷ L
Nu
is
hL
k
:
multiply
it
by
k
,
then
divide
by
L
.
h
=
80.7197
×
0.028
0.5
h
=
2.26
0.5
h
=
4.5203
W/(
m
2
·K)
Check
Put the numbers back in. Both sides should match.
3.50357 ×
10
8
=
9.81
×
40
×
0.5
3
320 × (1.75 ×
10
-5
) × (2.5 ×
10
-5
)
✓
80.7197
=
0.59
×
(3.50357 ×
10
8
)
0.25
✓
4.5203
=
80.7197
×
0.028
0.5
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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