lecture 14 heat - ernetaghosh/teaching/lecture14_heat.pdf · 2nd order differential equation: need...
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Lecture 14
Heat
1
EPS 122: Lecture 18 – Heat sources and flow
Heat sources and flow Reading: Fowler Ch 7
EPS 122: Lecture 18 – Heat sources and flow
Heat sources Sun
From the sun •� 2 x 1017 W •� 4 x 102 Wm-2
From the Earth’s interior •� 4.4 x 1013 W •� 8.7 x 10-2 Wm-2
Earth
Drives surface processes
•� water cycle
•� rain •� erosion
•� biosphere Drives deep Earth processes
•� convection •� plate
tectonics
•� intrusion •� metamorphism
•� volcanism
Earthquakes: 1011 W
Richard Allen lecture notes
3 types of heat transfer:
- conduction - convection - radiation
Pollack et al., 1993
Hasterock, 2010
Heat Flow Measurements - Edward Bullard pioneered the
method in 1930’s - Deep boreholes enable temperature
measurements avoiding influence of surface temperature variations
- Bullard’s penetrator - Characteristic heat flows of certain
geological units Axel Hagermann Phil. Trans. R. Soc. A 2005
Pollack et al. (1993) Source: Peter Bird
T
T + δT z + δz
z
N(t) =C1
t
N(t) =C1
(C2 + t)p
t =2
V1
rh
21 +
x
2
4
t =x
V2+
2h1
qV
22 �V
21
V1V2
drdr
=�GMr(r)r
2F
Q =�2pE
T
dE
dt
E =�2pE
T
dE
dt
Q =�k
DT
d
2
k è thermal conductivity: - physical ability of a material to conduct heat - Q measured in Watts/m2
- k measured in Wm-1oC-1
Conductive heat flow
Q(z) =�k
∂T
∂z
∂T
∂t
=k
rC
p
—2T +
A
rC
p
∂T
∂t
=k
rC
p
—2T +
A
rC
p
�u ·—T
∂2T
∂z
2 =�A
k
Q =�k
∂T
∂z
=�Q0
T =� A
2k
z
2 +(Qd +Ad)
k
z
3
z
z + δz
a Q(z)
a Q(z+δz)
Specific heat: Amount of heat necessary to raise the temperature of 1 kg of material by 1oC
N(t) =C1
t
N(t) =C1
(C2 + t)p
t =2
V1
rh
21 +
x
2
4
t =x
V2+
2h1
qV
22 �V
21
V1V2
drdr
=�GMr(r)r
2F
Q =�2pE
T
dE
dt
E =�2pE
T
dE
dt
Q =�k
DT
d
∂T
∂t
=k
rC
p
∂2T
∂z
2 +A
rC
p
2
1-D heat conduction equation:
∂T
∂t
=k
rC
p
—2T +
A
rC
p
3
3-D heat conduction equation:
κ èthermal diffusivity
Diffusion equation: ∂T
∂t
=k
rC
p
—2T +
A
rC
p
3
∂T
∂t
=k
rC
p
—2T +
A
rC
p
∂T
∂t
=k
rC
p
—2T +
A
rC
p
�u ·—T
3
advection term
Advection-diffusion equation:
Radioactive Heat Generation: the A-term
- Heat produced due to decay of radioactive isotopes
- U, Th, K most common radioactive elements on Earth
- Granite has greater radioactive heat generation
- Still only ⅕ of the total heat comes from the crust as there is so much more mantle
- Radioactive isotopes producing most heat are 238U,
235U, 232Th, 40K - Radioactive elements more abundant in the past
Geotherms - Temperature-depth profiles within the Earth
- Constant heat flux à steady-state temperature: - equilibrium geotherm
: 2nd order differential equation, needs 2 boundary conditions to solve
∂T
∂t
=k
rC
p
—2T +
A
rC
p
∂T
∂t
=k
rC
p
—2T +
A
rC
p
�u ·—T
∂2T
∂z
2 =�A
k
3
(i) T = 0 on z = 0
(ii) on z = 0
(i) T = 0 on z = 0 (ii) Q = -Qd on z = d
∂T
∂t
=k
rC
p
—2T +
A
rC
p
∂T
∂t
=k
rC
p
—2T +
A
rC
p
�u ·—T
∂2T
∂z
2 =�A
k
Q =�k
∂T
∂z
=�Q0
3
Q(z) =�k
∂T
∂z
∂T
∂t
=k
rC
p
—2T +
A
rC
p
∂T
∂t
=k
rC
p
—2T +
A
rC
p
�u ·—T
∂2T
∂z
2 =�A
k
Q =�k
∂T
∂z
=�Q0
T =� A
2k
z
2 +(Q
d
+Ad)
k
z
3
5
EPS 122: Lecture 18 – Heat sources and flow
Radioactive heat generation – the A term
Radioactive elements were more abundant
� they were e�t more abundant remember
EPS 122: Lecture 18 – Heat sources and flow
Equilibrium geotherms
�
2nd order differential equation: need two boundary conditions to solve
1. T = 0 at z = 0
2. Q = = -Q0 at z = 0 � solve
Define surface temperature and heat flow
Or, define T at surface and heat flow at some depth
1. T = 0 at z = 0
2. Q = -Qd at z = d � solve