SME
Heat Transfer · Problem type
Lumped cooling and the Biot check
Use this for “A 10 mm steel ball (ρ = 7800 kg/m³, c = 450 J/(kg·K), k = 40 W/(m·K)) at 300 °C cools in 25 °C air with h = 50 W/(m²·K). What is its temperature after 120 s?”
Chart
T(120) ≈ 123.6
Level T∞ = 25 never meets the curve
This is an example. Type your own values to start a new problem.
Bi
=
h (W/(m²·K))
×
(
D (m)
÷
6)
÷
k (W/(m·K))
τ (s)
=
ρ (kg/m³)
×
c (J/(kg·K))
×
(D
÷
6)
÷
h
T (°C)
=
T∞ (°C)
+
(
Tᵢ (°C)
−
T∞)
×
e
−
s
/τ
Formulas
Bi
=
h
×
D
÷ 6
k
0.002083
=
50
×
0.01 ÷ 6
40
τ
=
ρ
×
c
×
D
÷ 6
h
117
=
7,800
×
450
×
0.01 ÷ 6
50
T
=
T∞
+
(Tᵢ
−
T∞)
×
e
−
t
÷
τ
123.6058
=
25
+
(300
−
25)
×
e
−120 ÷ 117
Assumptions
•
The Biot number is under 0.1, so the body stays at one temperature inside as it cools.
•
A sphere of diameter D, so its volume over its surface is V ÷ A = D ÷ 6.
•
h, ρ, c and k stay the same while it cools; the air stays at T∞.
Step-by-step
WE KNOW
t = 120 s, D = 0.01 m, h = 50 W/(m²·K), k = 40 W/(m·K), ρ = 7,800 kg/m³, c = 450 J/(kg·K), Tᵢ = 300 °C, T∞ = 25 °C
FIND
Biot number (Bi), time constant (τ), temperature at t (T)
Step 1 · Find Biot number (Bi)
Use Bi = h × (D ÷ 6) ÷ k
Bi
compares
the
resistance
inside
the
body,
V ÷ A
k
,
with
the
film’s
1
h
.
Bi
=
50
×
0.01 ÷ 6
40
Bi
=
50
×
0.0016667
40
Bi
=
0.083335
40
Bi = 0.002083
Step 2 · Find time constant (τ)
Use τ = ρ × c × (D ÷ 6) ÷ h
The
heat
stored
per
degree,
ρc(
V
A
),
over
what
the
film
carries
off
per
degree,
h
.
τ
=
7,800
×
450
×
0.01 ÷ 6
50
τ
=
7,800
×
450
×
0.001667
50
τ
=
3,510,000
×
0.001667
50
τ
=
5,851.17
50
τ
=
117
s
Step 3 · Find temperature at t (T)
Use T = T∞ + (Tᵢ − T∞) × e−t ÷ τ
The
gap
to
the
air
shrinks
by
the
factor
e
−
t
÷
τ
;
add
what
is
left
of
it
to
T∞.
T
=
25
+
(300
−
25)
×
e
−120 ÷ 117
T
=
25
+
275
×
e
−
120
117
T
=
25
+
275
×
0.3586
T
=
25
+
98.615
T
=
123.6058
°C
Check
Put the numbers back in. Both sides should match.
0.002083
=
50
×
0.01 ÷ 6
40
✓
117
=
7,800
×
450
×
0.01 ÷ 6
50
✓
123.6058
=
25
+
(300
−
25)
×
e
−120 ÷ 117
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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