SME
Geophysics · Problem type
Temperature with radiogenic heat
Use this for “Crust with a surface heat flow of 62.5 mW/m², a conductivity of 2.5 W/(m·K) and 1 μW/m³ of radiogenic heat is 10 °C at the surface. How hot is it 10 km down?”
Chart
T(z) = −0.2z² + 25z + 10
The vertical axis grows downward.
This is an example. Type your own values to start a new problem.
q₀
Surface heat flow
example
k
Thermal conductivity
example · W/(m·K)
A
Heat production
example · μW/m³
T₀
Surface temperature
example · °C
z
Depth
example · km
T
Temperature at depth
calculated · °C
Formulas
T
=
T₀
+
q₀
×
z
k
−
A
×
z
2
2 ×
k
240
=
10
+
62.5
×
10
2.5
−
1 ×
10
2
2 × 2.5
Assumptions
•
Steady conduction; uranium, thorium and potassium in the rock make A microwatts of heat in each cubic meter, evenly.
•
The heat made above depth z leaves through the surface, so less flows up through z and the geotherm curves.
•
The rule holds while heat still flows up at z: Az stays under q₀ (μW/m³ × km is mW/m²).
Step-by-step
WE KNOW
q₀ = 62.5 mW/m², k = 2.5 W/(m·K), A = 1 μW/m³, T₀ = 10 °C, z = 10 km
FIND
temperature at depth (T)
Step 1 · Find temperature at depth (T)
Use T = T₀ + q₀ × z ÷ k − A × z² ÷ (2 × k)
Add
the
straight
warming
q₀
z
k
,
then
take
off
Az
²
2
k
for
the
heat
made
on
the
way
down.
T
=
10
+
62.5
×
10
2.5
−
10
2
2 × 2.5
T
=
10
+
62.5
×
10
2.5
−
10
2
5
T
=
10
+
62.5
×
10
2.5
−
100
5
T
=
10
+
625
2.5
−
20
T
=
10
+
250
−
20
T
=
240
°C
Check
Put the numbers back in. Both sides should match.
240
=
10
+
62.5
×
10
2.5
−
10
2
2 × 2.5
✓
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