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Unformatted text preview: 226 7 Heat _________—___—_________ Table 7.2. Relative abundance of isotopes and crustal heat generation in
the past relative to the present Relative abundance Heat generation Age
(Ma) “3U “5U U“ Th K Model A” Model B‘
Present [.00 1.00 1.00 1.00 1.00 1.00 1.00
500 1.08 1.62 1.10 1.03 1.31 1.13 1.17
1000 1.17 2.64 1.23 1.05 1.70 1.28 1.37
1500 1.26 4.30 1.39 1.08 2.34 1.48 1.64
2000 1.36 6.99 1.59 1.10 2.91 1.74 1.98
2500 1.47 11.4 1.88 1.13 3.79 2.08 2.43
3000 1.59 18.5 2.29 1.16 4.90 2.52 3601
3500 1.71 29.9 2.88 1.19 6.42 3.13 3.81 "This assumes a present isotopic composition of99.2886% 238U and 0.7114“,J 235U.
I'Model A based on Th/U = 4, K/U = 20,000. ‘Model B based on Th/U = 4, K/U = 40,000. Source: Jessop and Lewis (1978). are in the ranges 150—260 ppm, 80—100ppb and 15725 ppb, respectively.
These abundances result in a total radioactive heat production for the
crust and mantle of 1.4727 x 10‘3W, with a best guess value of 2.1 x10‘3W. 7.3 Calculation of Simple Geotherms 7.3.1 Equilibrium Geotherms As can be seen from Eq. 7.18, the temperature in a column of rock is
controlled by several parameters, some internal and some external to the
rock column. Internal parameters are the conductivity, speciﬁc heat,
density and radioactive heat generation. External factors include heat ﬂow
into the column, the surface temperature and the rate at which material
is removed from or added to the top of the column (erosion or deposition).
Temperature—depth proﬁles within the earth are called geatherms. If we
consider a onedimensional column with no erosion or deposition and a
constant heat ﬂux, the column may eventually reach a state of thermal
equilibrium in which the temperature at any point is steady. In that case,
the temperatureﬁdepth proﬁle is called an equilibrium geotherm. In this
equilibrium situation, BT/dt = 0, and Eq. 7.16 applies:
2
a—T = 7 5 (7.20)
v 622 k Since this is a secondorder differential equation, it can be solved given
two boundary conditions. Assume that the surface is at z = O and that 2
increases downwards. Let us consider two pairs of boundary conditions.
One possible pair is 7.3 Calculation of Simple Geotherms 227 ——_——_*_— (i) T=Oon z=0and
(ii) a surface heat flow Q = — kaT/dz = — QO on 2 =0. The surface heat ﬂow Q = —Q0 is negative because heat is assumed to be
ﬂowmg 'upwards out of the medium and z is positive downwards.
Integrating Eq. 7.20 once gives ~= ——+cl (7.21) where cl is the constant of integration. Because 37702 = Qu/k on 2 = 0 is
boundary condition (ii), the constant c1 is given by =%
k Substituting Eq. 7.22 into Eq. 7.21 and then integrating the second time
gives Cl (7.22) A Q
— __ 2 _0
l 2k2 + k z + £2 (7.23)
where cz is the constant of integration. However, since 7 = 0 on 2 = 0 was speciﬁed as boundary condition (i), c2 must equal zero. The temperature
Within the column is therefore given by __ A Qo
7".— —i22 + 72 (724) An alternative pair of boundary conditions could be (i) T=0 on 2:0 and
(ii) Q=—Qd on z=d. This could, for example, be used to estimate equilibrium crustal geotherms
if d was the depth of the crust/mantle boundary and Qd was the mantle
heat ﬂow into the base of the crust. For these boundary conditions
integrating Eq. 7.20 gives, as before, , (3T A E = —~];z + Cl (7.25) where {:1 is the constant of integration Because 6T 6 '
i _ . z = k on =
boundary condition (ii), c1 is given by / Qd/ Z d IS _ Qd Ad
_ 7 + 7 (up) Ci
Substituting Eq. 7.26 into Eq. 7.25 and then integrating again gives _ A Q +Ad
T— —Ezz+( dk )2“; (7.27) where c2 is theconstant of integration. Because T=D on z=0 was
boundary condition (1), c2 must equal zero. The temperature in the column Temperature (C)
D 500 1000 1500 N
0 Depth (km) b
D ea db: Figure 7.3. Equilibrium geotherms
calculated from Eq. 7.28 for 8 50km thick
column of rock. Curve a: standard model
with conductivity 2.5 W m"1 °C' 1,
radioactive heat generation 1.25pW m'
and basal heat flow 21 x10"’Wm“1.
Curve b: standard model with
conductivity reduced to 1.7 W m’ ‘ “C‘ ‘.
Curve c: standard model with radioactive
heat generation increased to 2.5 [1w rn ’ 3.
Curve d: standard model with basal heat
flow increased to 42 x 10‘ 3 W m'z.
Curve e: standard model with basal heat
flow reduced to 10.5 x10" Wm". (From Nisbet and Fowler 1982.) 3 228 7 Heat ____________—_————— 0 g z < d is therefore given by T: A 2 +(Qd+ Ad); _ _ 7.28
Zkz k ( ) Comparison of the second term in Eqs. 7.24 and 7.28 shows that a column
of material of thickness d and radioactive heat generation A makes a
contribution to the surface heat flow of Ad. Similarly, the mantle heat ﬂow
Qd contributes de/k to the temperature at depth 2. 7.3.2 OneLayer Models Figure 7.3 illustrates how the equilibrium geotherm for a model rock
column changes when the conductivity, radioactive heat generation and
basal heat flow are varied. This model column is 50 km thick, has
conductivity 2.5Wm'1 °C'1, radioactive heat generation 1.25;1Wm'3
and a heat flow into the base of the column of 21 x10‘3Wm'2.
The equilibrium geotherm for this model column is given by Eq. 7.28
and is shown in Figure 7.33; at shallow levels the gradient is
approximately 30°Ckm'1, whereas at deep levels the gradient is 15°C km"1 or less. Conductivity Reducing the conductivity of the whole column to
1.7Wm’l "C.1 has the effect of increasing the shallowlevel gradient to
about 45°C krri'l (see Fig. 7.3b). Increasing the conductivity to
3.4Wm'1 "C’1 would have the opposite eflect of reducing the gradient to about 23°Ckm'1 at shallow levels. Heat Generation Increasing the heat generation from 1.25 iiW m’3 to
2.5 [.iW m’ 3 raises the shallowlevel gradient to over 50°C km"1 (Fig. 7.30);
in contrast, reducing the heat generation to 0.4 ,uW m' 3 reduces this shallowlevel gradient to about 16°C km’ 1. Basal Heat Flow If the basal heat ﬂow is doubled from 21 to
42 ><10‘3Wm'2 the gradient at shallow level is increased to about
40°Ckm”1 (Fig.7.3d). If the basal heat flow is halved to 10.5X
IO'JWm“2, the shallowlevel gradient is reduced to about 27°C km‘l (Fig. 7.3e). 7.3.3 TwoLayer Models The models described so far have been very simple with a 50km thick
surface layer of uniform composition. This is not appropriate for the real
earth but is a mathematically simple illustration. More realistic models
have a layered crust with the heat generation concentrated towards the
top (see, e.g., Sect. 7.6.1). The equilibrium geotherm for such models is
calculated exactly as described in Eqs. 7.20—7.28 except that each layer
must be considered separately and temperature and temperature gradients
must be matched across the boundaries.
Consider a two—layer model: 7.3 Calculation of Simple Geotherms 229
A = A1 for 0 s z < 21
A=A2 for 21<z<22,
T = O on 2 = 0
with a basal heat flow Q = — Q2 on 2 = 22. In the first layer, 0 s z < 21, the
equilibrium heat conduction equation is
62 T _ A 1 In the second layer, 21 s z < 22, the equilibrium heat conduction equation
is
62 _ A:
622 k
The solution to these two diﬂ'erential equations, subject to the boundary conditions and matching both temperature T and temperature gradient
6T/dz on the boundary 2 = 21, is (7.30) _~A,2 Q2 A2 A2 T— ﬂz +[T+T(22—z,)+ L1} for O<z<z1 (7.31)
A; 2 Q2 A222 A —A T 2k2 +[k+ k z+[ 12k 2}} for 21<z<22 (7.32) Figure 7.4 shows an equilibrium geotherm calculated for a model
Archaean crust. The implication is that Archaean crustal temperatures
may have been relatively high (compare with Fig. 7.3.). 7.3.4 The TimeScale of Conductive Heat Flow Geological structures such as young mountain belts are not usually in
thermal equilibrium because the thermal conductivity of rock is so low
that equilibrium takes many millions of years to attain. For example
consrder the model rock column with the geotherm shown in Figure 7 3a,
If the _basal heat ﬂow were suddenly increased from 21 I to
42 x 10 3 Wm”, the temperature of the column would increase until
the new'equilibrium temperatures (Fig. 7.3d) were attained. That this
process is very slow can be illustrated by considering a rock at depth
20 km. The initial temperature at 20 km would be 567°C, and 20 Ma after
the basal heat flow increased, conduction would have raised the
temperature at 20km to about 580°C. Only after 100 Ma would the
temperature at 20 km be over 700°C and close to the new equilibrium
value of 734°C. This can be estimated quantitatively from Eq. 7.17: 87‘ BIT
_ = K ﬁ
61‘ 022
The characteristic time 1 = lz/K gives an indication of the amount of time necessary for a temperature change to propagate a distance of I in a
medium haVing thermal diffusivity ic. Likewise, the characteristic thermal Tennerature ('0)
o son 1000 150i: CRUST l MANTLE O .63 Figure 7.4. Twolayer model Archaen
crust and equilibrium geotherm. Heat
generation A in nWm”; heat flow from
mantle Q in 10‘3 Wm”. (After Nisbet
and Fowler 1982.) ,L W‘s(96b
7t ‘0  g
. v\ I. ,
K [9 WWI 9" iv“) ‘5“ ﬁr
LP W \o" 34%,“ “Viw It » .r—P“ ...
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 Fall '07
 ANANDAKRISHNAN
 Thermodynamics, Heat, Heat Transfer, heat generation

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