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Unformatted text preview: Heat 7.1 Introduction alcanoes, intrusions, earthquakes, mountain building and meta
orphism are all controlled by the transfer and generation of heat. The
rth’s thermal budget controls the activity of lithosphere and
.thenosphere as well as the development of the innermost structure of
.e earth. Heat arrives at the earth’s surface from its interior and from the sun.
irtually all the heat comes from the sun, as any sunbather knows, but
all eventually radiated back into space. The rate at which heat is received
y the earth, and reradiated, is about 2 x 1017W or, averaged over the
lrface, about 4 X 102 Wm'z. Compare this value with the mean rate of
15s of internal heat from the earth, 4 x 1013 W (or 8 X 10'2 Wm'z); the
)proximate rate at which energy is released by earthquakes, 1011 W; the
rte at which heat is lost by a clothed human body on a very cold (— 30°C),
indy (10 m s‘ 1) Canadian winter day, 2 x 103 W m”. From a geological
erspective, the sun’s heat is important because it drives the surface water
/cle, the rainfall and, hence, erosion. However, the heat source for igneous
1trusion, metamorphism and tectonics is in the earth, and it is this internal
)urce which accounts for most geological phenomena. The sun and the
iosphere have kept the surface temperature within the range of the
:ability of liquid water, probably 15—25°C averaged over geological time.
iiven that constraint, the movement of heat derived from the interior has
overned the geological evolution of the earth, controlling plate tectonics,
gneous activity, metamorphism, the evolution of the core and hence the
arth’s magnetic ﬁeld. Heat moves by conduction, convection, radiation and advection.
‘onduction is the transfer of heat through a material by atomic or
rolecular interaction within the material. In convection, heat transfer
uccurs because the molecules themselves are able to move from one
)cation to another within the material; it is important in liquids and
;ases. In a room with a hot lire, air currents are set up which move the
ight, hot air upwards and away from the fire while dense cold air moves
[1. Convection is a much faster way of transferring heat than conduction.
\s an example, when we boil a pan of water on the stove, the heat is
ransferred through the metal saucepan by conduction but through the
vater primarily by convection. Radiation involves direct transfer of heat 220 7 Heat by electromagnetic radiation (e.g., from the sun or an electric bar heater).
Within the earth, heat moves predominantly by conduction through the
lithosphere (both oceanic and continental) and the solid inner core.
Although convection cannot take place in rigid solids, over geological
times the earth’s mantle appears to behave as a very highviscosity liquid,
which means that slow convection is possible in the mantle (see Sects. 6.1,
7.4 and 7.8); in fact, heat is generally thought to be transferred by
convection through most of the mantle as well as through the liquid outer
core. Although hot lava radiates heat, as do crystals at deep, hot levels
in the mantle, radiation is a minor factor in the transfer of heat within
the earth. Advection is a special form of convection. If a hot region is
uplifted by tectonic events or by erosion and isostatic rebound, heat (called
advected heat) is physically lifted up with the rocks. It is not possible to directly measure temperatures deep in the earth.
Temperatures and temperature gradients can only be measured close to
the earth‘s surface, usually in boreholes or mines or in oceanic sediments.
The deeper thermal structure must be deduced by extrapolation, by
inference from seismic observations, from knowledge of the behaviour of
materials at high temperatures and pressures, from metamorphic rocks
and from models of the distribution of heat production and of the earth’s
thermal evolution. 7.2 Conductive Heat Flow 7.2.1 The Heat Conduction Equation Heat, as everyone knows, flows from a hot body to a cold body, and not
the other way around. The rate at which heat is conducted through a
solid is proportional to the temperature gradient (the difference in
temperature per unit length). Heat is conducted faster when there is a
large temperature gradient than when there is a small temperature gradient
(all other things remaining constant). Imagine an infinitely long and wide
solid plate, 11 in thickness, with its IOWer surface kept at temperature T,
and its upper surface at temperature T2 (T2 > T1). The rate of flow of heat
per unit area down through the plate is proportional to T,— T, 7.1
d ( ) The rate of ﬂow of heat per unit area up through the plate, Q, is therefore _,ZL:£
Q_k<d) where k, the constant of proportionality, is called the thermal conductivity.
The thermal conductivity is a physical property of the material of which
the plate is made and is a measure of its physical ability to conduct
heat. The rate of flow of heat per unit area Q is measured in units 01'
watts per square metre (Wm‘z), and thermal conductivity k is mea (7.2) i 1..1.,,,mhz»it.~a i .. ‘vIL’uV' zit—1.112!“ , i .W'mazejs (a: 7.2 Conductive Heat Flow 22] sured in watts per metre per degree centigrade (Wm'1 °C'1).* Values of
the thermal conductivity of solids vary widely: 418 W m’1 °C' 1 for silver;
159Wm'I "C‘1 for magnesium; 1.2Wm'l °C‘ ‘ for glass; 1.7—3.3 Wm—1
°C"1 for rock; 0.1 Wm" 1 °C"1 for wood. To express Eq. 7.2 as a differential equation, let us assume that the
temperature of the lower surface (at z) is T and that the temperature of
the upper surface (at z + 62) is T + 6T (Fig. 7.1). Substituting these values into Eq. 7.2 then gives
T + 6T — T
Q(2) =  4‘) (52
(ST
= — k — 7.3
62 ( )
In the limit as 52—»0, Eq. 7.3 is written 6T
Q(z) =  k f (74) 82 The minus sign in Eq. 7.4 arises because the temperature is increasing in
the positive 2 direction; since heat flows from a hot region to a cold region,
it flows in the negative 2 direction. If we consider Eq. 7.4 in the context of the earth, 2 denotes depth
beneath the surface. As 2 increases downwards, a positive temperature
gradient (temperature increases with depth) means that there is a net ﬂow
of heat upwards out of the earth. Measurement of temperature gradients
and conductivities in nearsurface boreholes and mines can provide
estimates of the rate of loss of heat from the earth. Consider a small volume element of height 62 and crosssectional area a (Fig. 7.2). Any change in temperature 6T of this small volume element
in time 6t depends on 1. flow of heat across the element’s surface (net flow is in or out),
2. heat generated in the element and
3. thermal capacity (specific heat) of the material. The heat per unit of time entering the element across its face at z is
aQ(z), whereas the heat per unit time leaving the element across its face
at z + 62 is aQ(z + 52). Expanding Q(z + 62) in a Taylor series gives Q(z+6z)=Q(z)+5za—[email protected]—Q+m 02 2 622 (75) In the Taylor series, the (62)2 term and those of higher order are very small
and can be ignored. From Eq. 7.5 the net gain of heat per unit time is "‘ Until recently, the c.g.s. system was used in heat flow work. In that system, lhgu (heat
generation unit): 10‘13calcm'ls’l 24.2 x10'7Wm'3; 1th (heat flow unit):
10“’calem‘zs‘l 24.2 x IOAZWm”; and conductivity, 0.006calcm"s'”C‘I =
252qu °C“. hot T+5T cold FLOW OF HEAT Figure 7.1. Conductive transfer of heat
through an inﬁnitely wide and long plate
dz in thickness. Heat flows from the hot
side of the slab to the cold side (in the
negative 2 direction). aQ(z+Sz) Figure 7.2. Volume element, height dz,
crosssectional area a. Heat is conducted
into and out of the element across the
shaded faces only. We assume there is no
heat transfer across the other four faces. 222 7 Heat
_______—_______________ heat entering across 2 — heat leaving across 2 + 62 = aQ(z) — aQ(z + 62)
3Q = —aoz—
dz (7.6) Suppose heat is generated in this volume element at a rate A per unit
volume per unit time. The total amount of heat generated per unit time is then An 62 (7.7) Radioactive heat is the main internal heat source for the earth as a whole; however, local heat sources and sinks include latent heat, Ishear heating and endothermic/exothermic chemical reactions. Radioactive heat generation is discussed in Section 7.2.2. Combining expressmns 7.6 and 7.7 gives the total gain in heat per unit time to ﬁrst order in 52 as
5Q 6—62—
Aaz 0 Bl The speciﬁc heat cl, of the material of which the element is mhade
determines the temperature rise due to this ga‘in‘in heat Since speczﬁc left
is deﬁned as the amount of heat necessary to raise a1 kglof the materia y
1°C. Speciﬁc heat is measured in units of Wkg C . If the material has density p and speciﬁc heat CD, and undergoes a
temperature increase (ST in time 61, the rate at which heat is gained 13 (ST
ovating (7.8) (79) Thus, equating the expressions 7.8 and 7.9 for the rate at which heat
is gained by the volume element gives 6
cpaongz Aaéz 7 1162 —Q dz cppg=A—§§ (7.10)
In the limiting case when 62,6t —>0, Eq. 7.10 becomes appZ—ZT=A—‘:vg (7.11)
Using Eq. 7.4 for Q (heat ﬂow per unit area), we can write c,,p%n=A+k%:—Zw (7.12)
or 6T_ k (VT 1 (7‘13) ——i#—+
at peptizz pcp This is the onedimensional heat conduction equation.
In the derivation of this equation, temperature was assumed to be a
function only of time t and depth 2. It was assumed not to vary in the x and y 2i
p
,x 7.2 Conductive Heat Flow 223
\
directions. If temperature were assumed to be a function of x, y, z and t, a
threedimensional heat conduction equation could be derived in the same
way as this onedimensional equation. It is not necessary to go through the
algebra again: We can generalize Eq. 7.13 to a threedimensional Cartesian
coordinate system as 6T k 627“ 627" (FT A
a—p—q}[a§+ay—Z+y]+kc—P (7.14)
Using differential operator notation (see Appendix 1), we write Eq. 7.14 as
T k A
L=—V2T+~— (7.15)
(it pcp pcl, Equations 7.14 and 7.15 are both known as the heat conduction equation.
The term k/pcp is known as the thermal diﬂusivily K. Thermal diffusivity
expresses the ability of a material to lose heat by conduction. Although
we have derived this equation for a Cartesian coordinate system, we can
use it in any other coordinate system (e.g., cylindrical or spherical) provided
we remember to use the deﬁnition of the Laplacian operator, V2 (Appendix
I), which is appropriate for the desired coordinate system. For a steadystate situation when there is no change in temperature
with time, Eq. 7.15 becomes A
V2T= — — (7.16)
k
In the absence of any heat generation, Eq. 7.15 becomes
6T k
VA=—VZT (7.17)
at pcp This is the diﬁhsion equation. So far we have assumed that there is no relative motion between the
small volume of material and its immediate surroundings. Now consider
how the temperature of the small volume changes with time if it is in
relative motion through a region where the temperature varies with depth.
This is an effect not previously considered, and so Eq. 7.13 and its
threedimensional analogue, Eq. 7.15, must be modiﬁed. Assume that the
volume element is moving with velocity uz in the z direction. It is now
no longer fixed at depth 2; instead, at any time t, its depth is z + uzt. The
BT/at in Eq. 7.13 must therefore be replaced by oT/az + (dz/dt)(6T/0z).
The first term is the variation of temperature with time at a ﬁxed depth
2 in the region. The second term (dz/dtJ(aT/az) is equal to uz(BT/62) and
accounts for the effect of the motion of the small volume of material through the region where the temperature varies with depth. Equations
7.13 and 7.15 become, respectively, 6T kﬂzT A 6T —=—* —— ‘ 7.1 a: )Jcpc"22_i—/_)cFl u‘az ( 8)
and T k a—=»V2T+iiu VT (119) at pcp pop 224 7 Heat In Eq. 7.19, n is the threedimensional velocity of the material. The term
u'VT is the advective transfer term. Relative motion between the small volume and its surroundings can
occur for various reasons. The difﬁculty involved in solving Eqs. 7.18 and
7.19 depends on the cause of this relative motion. If material is being
eroded from above the small volume or deposited on top of it, then the
volume is getting nearer to or farther from the cool surface of the earth.
In these cases, u, is the rate at which erosion or deposition is taking place.
This is the process of advection referred to earlier. On the other hand, the
volume element may form part of a thermal convection cell driven by
temperatureinduced density differences. In this latter case, the value of
142 depends on the temperature ﬁeld itself rather than on an external factor
such as erosion rates. The fact that, for convection, u, is a function of
temperature makes Eqs. 7.18 and 7.19 nonlinear and signiﬁcantly more diilicult to solve. 7.2.2 Radioactive Heat Generation Heat is produced by the decay of radioactive isotopes (Table 6.2). Those
radioactive elements which contribute most to the internal heat generation
of the earth are uranium, thorium and potassium. These elements are
present in the crust in very small quantities, parts per million for uranium
and thorium and of the order of a percent for potassium; in the mantle
they are some two orders of magnitude less abundant. Nevertheless, these
radioactive elements are important in determining the temperature and
tectonic history of the earth. Other radioactive isotopes, such as aluminium
26 and plutonium 244, may have been important in the earliest history of the planet.
The radioactive isotopes producing most of the heat generation in the crust are 233U, 235U, 232Th and 40K. The uranium in the crust can be
considered to be 233U and 235U, with presentday relative abundances of
99.28% and 0.72%, respectively; but “K is only present at a level 1:10‘
of total potassium (Chapter 6). Table 7.1 gives the radioactive heat
generation of some average rock types. It is clear from this table that, on
average, the uranium and thorium contributions to heat production are
larger than the potassium contribution. On average, granite has a greater
internal heat generation than maﬁc igneous rocks, and the heat generation
of undepleted mantle is very low. The heat generated by these radioactive isotopes as measured today
can be used to calculate the heat generated at earlier times. At time t ago,
a radioactive isotope with a decay constant ,1 would have been a factor
e“ more abundant than today (Eq. 6.5). Table 7.2 shows the changes in
abundance of isotopes and consequent higher heat generation in the past
relative to the present. Although the heat generation of the crust is some two orders of
magnitude greater than that of the mantle, the rate at which the earth as
a whole produces heat is influenced by the mantle because the volume of
the mantle is so much greater than the total crustal volume. About oneﬁfth
of radioactive heat is generated in the crust. The mean abundances of
potassium, thorium and uranium, for the crust and mantle taken together, i} mum";  .=« Table 7.1. ‘ ' ' ' '
Typical concentrations of radioactive elements and heat production of some rock types Average
oceanic
crust Average
continental
upper crust Undepleted
mantle Alkali
basalt Peridotite Tholeiitic
basalt Granite 002
0.10
002 09
2.7
04 16
5.8
20 0 006
0.04
O 01 0.8
2 5
1.2 01
0.4
02 4
15
35 a
.:
E’
Q)
a
A
I)
2
.9.
a
E A
age
SQEA
gez§
UDHM Heat generation (10‘1" Wkg“) 0 02
0 03
0.007
0 057 1.7
2.9 0 9
0.7
0 1
0.5 16
1.6 0.7
3.9 2.7 0 006
0.010
0 004
0 020 1.9 08
0.7
04
2.7 01
01
0.1
03 3.9 4.1
1.3 9.3 U Th K
Total 3.2
0.02 3.2 2.8 2.7
2.5 Density (103 kgm‘ 3) 1.0 0.006 0.5 0.08 Heat generation (,qu‘ 3) 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 ...
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 Fall '07
 ANANDAKRISHNAN
 Heat, Heat Transfer, heat conduction equation, heat generation

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