EES215

Lecture 7

Heat flow from the earth: potential energy sources: heat generation; rotation; gravity

heat generating nuclei: U-238; U-235; Th-232; K-40
characteristics of these nuclei; distribution in earth

Radioactivity:

 

Radioactive decay: characteristic of nucleus of given isotope, independent of temperature and pressure.

Decay equation: N (t) = N (0) e-lt with the following definitions:

N (t); N(0) number of atoms at time t and time 0, respectively

l decay constant, characteristic of given isotope

 

Half-life T1/2 = ln 2/l

 

Three common types of radioactivity:

a-decay: 238U à 234Th + 4He + Q;   4He is called an a-particle here; Q is energy released by this reaction

b-decay: 234Th à 234Pa + e- + Q;  e- is the b-particle; in addition a particle called anti-neutrino is released in this reaction

g-decay: 234Pa* à 234Pa + Q; exited state steps down to ground state releasing a photon or g-particle

The above examples are part of the decay chain reactions of 238U

 

Four important reactions for heat production in the earth:

 

238U à 206Pb + 8 4He + 6 e- + Q  T1/2 = 4.468 Ga

235U à 207Pb + 7 4He + 4 e- + Q  T1/2 = 0.704 Ga

232Th à 208Pb + 6 4He + 4 e- + Q  T1/2 = 14.01 Ga

40K à 40Ar + Q  or 40Ca + e- +Q  T1/2 = 1.250 Ga

(branched decay, EC = electron capture and b-decay)

 

 

 

Heat flow from the earth: Heat flow: transfer of energy from hot to cold, from interior to surface of the earth

Heat flow at the surface of the earth:  0.06 W/m2 = 60 x 10-3 W/m2

Potential energy sources: heat generation; minor contributions from rotation and gravity

Heat generating nuclei: U-238; U-235; Th-232; K-40
characteristics of these nuclei; distribution in earth
 
 

Table 1. Distribution of heat producing elements in different materials

Material

U, ppm

Th, ppm

K, %

Mantle

0.026

0.103

0.026

Basalt 

0.1

0.35

0.2

Granite

4

17

3.2


 

Table 2. Half lives and energy output of radioactive isotopes

Isotope

Half life,
years

Power (isotope),
mW/kg

Power (element),
mW/kg

Total heat in
Earth, 1012 W

238U

4.468 x 109

95.0

94.35

11.33

235U

7.038 x 108

562.0

4.05

0.486

U (total)

 

 

98.4

 

232Th

1.4 x 1010

26.6

26.6

11.48

  40K

1.25 x 109

30.0

0.0035

8.41


Heat flow: transfer of energy from hot to cold, from interior to surface of the earth

Mechanism of heat transfer:
Conduction: heat exists as vibrations of atoms, intensity of vibration determines temperature; vibrational energy is transferred from one particle to the next; heat is transferred along temperature gradient. Quantity of heat transferred per unit time is proportional to temperature difference and to thermal conductivity: function of density, crystal lattice, good thermal conductors are in general also good electric conductors (metals). Conduction is an important, but slow process.

Important heat flow equations:

Conduction:

Q = - K T/z  (finite difference Q= - K DT/Dz ) (K is thermal conductivity; minus sign is convention since depth is measured from surface)

Temperature distribution in formation with surface heat flow Q0, heat production A, thermal conductivity K and surface temperature To:
T (z) = -(A/2K)z2 + (Q0/K)z + To

Heat flow: units: 10-3 W/m2 = 0.0239 HFU;
1 HFU (heat flow unit) = 10-6cal-cm-2-s-1 (older unit, still found in many papers)
1 HGU (heat generation unit) = 10-13 cal cm-1 s-1 oC-1 = 4.19x10-7 W/m3

 

Examples:               


Sandstone 1.5-4.2 W/m-deg          
Gneiss 2.1-4.2
Basalt 1.3-2.9
Granite 2.4-3.8


Halite 5.4-7.2           

Fe 73
Ag 417


Fiberglass 0.013

Wood 0.18