FowlerHeat1 - Heat 7.1 Introduction alcanoes, intrusions,...

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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 high-viscosity 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 (7-4) 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 near-surface 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 cross-sectional 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—Q+@az—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, cross-sectional 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. Radio-active 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 one-dimensional 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 three-dimensional heat conduction equation could be derived in the same way as this one-dimensional equation. It is not necessary to go through the algebra again: We can generalize Eq. 7.13 to a three-dimensional 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 steady-state 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 three-dimensional 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 three-dimensional 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 temperature-induced 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 present-day 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 one-fifth 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 one-dimensional 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 second-order 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 (7-24) 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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This note was uploaded on 07/23/2008 for the course GEOSC 203 taught by Professor Anandakrishnan during the Fall '07 term at Pennsylvania State University, University Park.

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FowlerHeat1 - Heat 7.1 Introduction alcanoes, intrusions,...

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