SME
Transport Phenomena · Problem type
Centre temperature of a wire with heat generation
Use this for “A 2 mm-radius rod with k = 20 W/(m·K) makes 1 × 10⁸ W/m³ of heat, and its surface is at 90 °C. How hot is its centre?”
Diagram
Tc
−
Tₛ
=
SR²
4k
=
5 ×
10
8
×
0.001
2
4 × 12
=
10.42
K:
the
heat
made
inside
must
flow
out,
so
the
centre
runs
hottest,
a
parabola
in
r.
This is an example. Type your own values to start a new problem.
Tc (°C)
−
Tₛ (°C)
=
S (W/m³)
×
(
R (m)
)
2
4
×
k (W/(m·K))
Formulas
Tc
−
Tₛ
=
S
×
R
2
4 ×
k
70.4167
−
60
=
5 ×
10
8
×
0.001
2
4 × 12
Assumptions
•
Steady radial conduction with heat made evenly through the wire; k is constant.
•
The surface is held at Tₛ; all the heat made inside leaves through it.
•
T(r) is a parabola, hottest at the centre: Tc − Tₛ = SR² ÷ 4k.
Step-by-step
WE KNOW
S = 5 × 10⁸ W/m³, R = 0.001 m, k = 12 W/(m·K), Tₛ = 60 °C
FIND
centre temperature (Tc)
Step 1 · Find centre temperature (Tc)
Use Tc − Tₛ = S × R² ÷ (4 × k)
The
heat
made
inside
must
flow
out,
so
the
centre
runs
SR
²
4
k
above
the
surface.
Tc
=
Tₛ
+
S
×
R
2
4 ×
k
Tc
=
60
+
(5
×
10
8
)
×
0.001
2
4 × 12
Tc
=
60
+
500
48
Tc = 70.4167 °C
Check
Put the numbers back in. Both sides should match.
70.4167
−
60
=
5 ×
10
8
×
0.001
2
4 × 12
✓
Topic check
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