Heat Flow - PowerPoint by wulinqing

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									Heat Flow




            1
                 Heat Flow
• Earth’s thermal budget controls:
  – Volcanoes and intrusives
  – Plate tectonic movement
  – Earthquakes
  – Mountain building
  – Metamorphism
  – Thermal structure of the Earth’s interior


                                                2
 Why do we care about heat flow?
• It is the fundamental cause of plate motions
• It is an excellent indicator of the age of the
  oceans
• It is a decent indicator of the age of the
  continents
• It is an indicator of tectonic activity
• It is related to uplift and subsidence via heating
  (expansion) and cooling (contraction)
• Heat is an inexhaustible source of energy but
  much new technology is needed (steam versus
  hot dry rock)

                                                       3
               Mean Rate of
        Heat Radiation Loss on Earth
Source                       Energy radiated       Averaged over
                                     (Watts)       Earth’s surface
                                                        (Watts/m2)
Solar                         2 x 1017 W              400 W/m2
Internal Heat                4.4 x 1013 W 8.7 x 10-2 W/m2
Energy released from              ~1011 W
Earthquakes

Heat loss from clothed                              2000 W/m2
Human body on cold,
(-30C) windy (10 m/s) day                     (note: frostbite occurs
(wind chill = -47C)                                    < 15 minutes)

                                                                         4
                              GEOS 4320                                  4
         Heat Flow Methods
• Conduction: transfer of heat through a
  material by atomic or molecular interaction
  within the material.
• Convection: transfer of heat by movement
  of molecules from one location to another
  within the material
• Water pot on stove: heat transferred
  through pot by conduction, through water
  by convection.
                                            5
                    GEOS 4320               5
         Heat Flow Methods
• Heat flow is faster via convection than
  conduction
• Radiation: direct transfer of heat by
  electromagnetic radiation (minor factor
  within the earth)
• Advection: heat physically lifted up with
  the rocks via tectonic events, erosion, and
  isostatic rebound.
                                                6
                    GEOS 4320                   6
       Conductive Heat Flow




• Heat flows from a hot body to a cold body
• rate of heat conduction through a solid 
  temperature gradient
• High heat flow  high temperature gradient
                                               7
                      GEOS 4320                7
         Conductive Heat Flow
•      Q(z) = -k[(T + dT – T)/dz] (7.3)
• where:
    Q   is the rate of flow per unit area down
      (positive z direction) (W m-2)
     k is the thermal conductivity of the material
      the heat is traveling through (W m-1 C-1)
     T is temperature (C)
     z is depth from surface (m) (positive
      downwards)
                                                      8
                         GEOS 4320                    8
        Conductive Heat Flow
• If take the limit as dz → 0 of Eqn. 7.3, then:

•          Q(z) = -k[dT/dz]                        (7.4)
• This is Fourier’s law of conduction: heat flow is
  proportional to the thermal gradient
• positive temperature gradient = net flow of heat
  upward (out of Earth)
• thus, positive temperature gradient means
  negative rate of flow of heat: Q
                                                       9
                        GEOS 4320                      9
 Measuring heat flow (Q) is simply a matter of
 dropping a thermometer down a drill hole and
 logging the change in temperature with depth

Q = -k T/z
k = thermal conductivity




                                                 10
Measuring Heat Flow
             Temperature
              readings are on the
              lithological
              boundaries
             Geothermal
              gradient and heat
              flow calculation
              values are placed at
              center-thickness of
              each unit




                                11
 Measurements of temperature gradients and thermal
conductivity in boreholes and mines provide estimates
      of the rate of loss of heat from the Earth




                                                        12
                       GEOS 4320                        12
      Thermal Conductivities
Substance                        k (W m-1 C-1)
Silver                                       418
Magnesium                                    159
Silica / a-Quartz                    1.36 / 7.69
Calcite                                     3.57
Olivine (Fa – Fo)                   3.16 – 5.06
“rock”                          1.7 – 3.3 (→ 6)
Glass                                         1.2
Wood                                          0.1
                                                13
                    GEOS 4320                   13
Conductive Heat Flow




                       14
        GEOS 4320      14
k=l




      15
Heat Conduction through a Volume
For a small volume of height dz and area a,
 any change in temperature dT in time dt
 depends on:
  1. the flow of heat across the volume’s
     surface (net flow is in or out)
  2. the heat generated in the volume
  3. the thermal capacity (specific heat) of
     the material
                                               16
                    GEOS 4320                  16
Heat Conduction through a Volume




                               17
              GEOS 4320        17
Heat Conduction through a Volume
• Specific Heat (cP) of a material is the
  amount of heat necessary to raise the
  temperature of 1kg of that material by 1C
• Specific heat is measured in J kg-1 C-1
  (J = Joules)
• Q = mcPT
• localized heat sources / sinks: latent heat,
  shear heating, and endothermic/
  exothermic chemical reactions

                                             18
                     GEOS 4320               18
             Specific Heats
Substance                       cP (kJ kg-1 C-1)
Silver
Magnesium
Silica / a-Quartz                   0.70 / 0.698
Calcite                                     .793
Olivine (Fa)                                0.55
“rock”
Glass
Wood
                                                19
                    GEOS 4320                   19
Heat Conduction through a Volume
• The net gain of heat per unit time is equal to [the
  heat entering across z] minus [the heat leaving
  across (z + dz)]:

•              aQ(z) – aQ(z + dz)
•                -adz[∂Q/∂z]                     (7.6)
        (applying the Taylor expansion series)
• Internal heat generation (e.g., radioactivity) at a
  rate A per unit volume per unit time is given as

•                      Aadz                      (7.7)
                                                     20
                        GEOS 4320                       20
Heat Conduction through a Volume
• Given that the material has a density r and
  specific heat cP, with a temperature increase dT
  in time dt (dT, dt → 0 ), then heat is gained at the
  rate:

•               cP adz r [∂T/∂t]               (7.9)

• combining and equating these expressions gives
  the rate at which heat is gained by the volume
  element:
• cP adz r [∂T/∂t] = Aadz - adz[∂Q/∂z]
                                                       21
                        GEOS 4320                      21
Heat Conduction through a Volume

• cP adz r [∂T/∂t] = Aadz - adz[∂Q/∂z]
• as adz is common to all three terms, it can be
  divided out, leaving:

•      cPr [∂T/∂t] = A - [∂Q/∂z]              (7.10)

• substituting for Q (heat flow / unit area) Eqn. 7.4,

•    cPr [∂T/∂t] = A + k[∂2T/∂z2]              (7.12)

                                                       22
                        GEOS 4320                      22
Heat Conduction through a Volume
•      cPr [∂T/∂t] = A + k[∂2T/∂z2]           (7.12)

• solving for ∂T/∂t gives the one-dimensional heat-
  conduction equation:

• [∂T/∂t] = (k/cPr)[∂2T/∂z2] + A/cPr
                                             (7.13)

• First derivatives indicate slope (temperature
  gradients)
• second derivatives indicate minima and maxima
  (heat sinks and sources)
                                                  23
                       GEOS 4320                   23
Heat Conduction through a Volume
• Expanding the second derivative term to three
  dimensions involves the Laplacian operator:
•        { [∂2T/∂x2] + [∂2T/∂y2] + [∂2T/∂z2] } = 2T
• Substituting 2T into Eqn. 7.13 gives the three-
  dimensional heat-conduction equation:

•      [∂T/∂t] = (k/cPr)2T + A/cPr
                                                (7.15)
• where k/cPr represents the thermal diffusivity k,
  the ability of a material to lose heat by
  conduction.            GEOS 4320
                                                    24
                                                    24
Heat Conduction through a Volume
• In a steady-state situation, dT/dt =0, and there is
  no change in temperature with time, then:
•              2T = -A/k                   (7.16)
• In the special situation where there is no heat
  generation (A = 0):
•             [∂T/∂t] = k2T                 (7.17)

• where k (= k/cPr) represents the thermal
  diffusivity, the ability of a material to lose heat by
  conduction. This is the diffusivity equation.
                                                       25
                         GEOS 4320                     25
Heat Conduction through a Volume
• Advection: heat physically lifted up with the
  rocks via tectonic events, erosion, and isostatic
  rebound.
• If we consider how the temperature of a small
  volume changes with time if it is relative motion;
  a additional term has to be added to the one-
  dimensional heat conduction equation.
• Let uz be the velocity of the moving volume
  element in the z direction; at any time t its depth
  is z + uzt. The expression uz(∂T/∂z) accounts for
  the affects of the volume motion through the
  thermal gradient.
                                                    26
                        GEOS 4320                   26
Heat Conduction through a Volume
• If we expand the velocity to three dimensions,
  then u = u(x, y, z) is the velocity vector of the
  material. The term u • 2T, the advective-
  transfer term, is added to the three-dimensional
  heat-conduction equation:

• [∂T/∂t] = k2T + A/cPr - u • 2T
                                              (7.19)
• Most commonly, uz is the rate at which erosion
  or deposition takes place; uz could also
  represent the rate of isostatic rebound.
                                                      27
                       GEOS 4320                      27
 Radiogenic heat production (A)
 Isotope      A, µW kg-1    H (W kg-1)       τ½ (a)      C (kg kg-1)
 238U                       9.46 x 10-5    4.47 x 109    30.8 x 10-9
 235U                       5.69 x 10-4    7.04 x 108    0.22 x 10-9
 Total U        96.7        9.91 x 10-5                  31.0 x 10-9
 232Th          26.3        2.64 x 10-5    1.40 x 1010    124 x 10-9
 40K                        2.92 x 10-5    1.25 x 109    36.9 x 10-9
 Total K       0.0035       4.48 x 10-9                  31.0 x 10-5


Table of main parameters for heat generating isotopes in the Earth. A
is the heat production of the element per unit weight; H is heat release
of the isotope or element per unit weight for the concentration C; t½ is
its half life in years; and C is the present mean mantle concentration of
the heat producing elements [Modified from Table 4-2, page 137, D.
L. Turcotte and G. Schubert, Geodynamics (2nd ed.), Cambridge
                                                                      28
University Press, 456 pp., 2002.]
               Important Parameters for
                 Heat Flow Analysis
Property, Symbol                    Approximate Range
Heat flow, Q                               0 - 125 mW m-2
Vertical temperature
gradient, dT/dz                            10 to 80 °C/km
Thermal conductivity, k                                   .
          - marine sediments           0.6 - 1.2 W m-1 °C-1
          - continental sediments          1 - 5 W m-1 °C-1
heat generation, A                         0-8 10-6 W m-3
Specific heat, cP                    0.85-1.25 kJ kg-1 °C-1
Density of crustal rocks
and lithosphere, r                    2200 to 3400 kg m-3

                                                         29
           Heat Flow Data for Continents




                                                         30
Artemieva and Mooney,JGR, 2001   Global heat flow measurements
Age crustal formation vs. Heat Flow
                   Plot of heat flow vs. tectonothermal age.
                   Data are plotted in the following age
                   groups: C – Cenozoic; M – Mesozoic;
                   LPa – Late Paleozoic; E – Early
                   Paleozoic; LPr – Late Proterozoic; E –
                   Early Proterozoic; A – Archean.
                   On the heat flow scale the plus sign
                   indicate the data mean with the box
                   height indicating +/- one standard
                   deviation of the data. On the age scale
                   the plus sign indicate the middle of the
                   age range and the box width indicates
                   the age range. Numbers in parentheses
                   indicates the number of data in each age
                   group. [Reproduced from Morgan, P.,
                   The thermal structure and thermal
                   evolution of the continental lithosphere,
                   Phys. Chem. Earth, 15, 107-193, 1984].
                                                      31
 Craton-edge
  Heat Flow
Measurements




 The Golden gate east of Clarens
                             32
Craton-edge Heat Flow
    Measurements




                        33
           Craton-edge Heat Flow
               Measurements




Why in the world is finding the edge of the craton important?
                                                            34
Kaapvaal Craton Geotherms




            The intersection of the
            geotherm with the adiabat
            indicates the thickness of
            the lithosphere



                                         35
Kaapvaal Craton Geotherms and Kimberlites




                                       36
Kaapvaal Craton Geotherms and Kimberlites


Letseng-le-Terae
Kimberlite




                                  The 603 ct
                                  Lesotho
                                  Promise
                                  Diamond


                                           37
     Calculation of Geotherms
• Temperature in a column of rock is controlled by
  several parameters:
  – Internal: conductivity, specific heat, density, and
    radioactive heat generation
  – External: heat flow into the column, surface
    temperature, erosion/deposition at the top of the
    column
• Equilibrium geotherm: column of rock (1-D) with
  no erosion/deposition and constant heat flow;
  column reaches a state of thermal equilibrium-
  temperature at any point is steady (∂T/∂t = 0).
                                                          38
          Calculation of Geotherms
•     Equilibrium geotherm: As ∂T/∂t = 0, Eqn 7.16
      applies:

•                    ∂2T/∂z2 = -A/k                    (7.20)
•     Boundary conditions:
    i.    T = 0 at z = 0
    ii.   surface heat flow Q = -k(∂T/∂z) = -Q0 at z = 0
•     Integrating Eqn 7.20 once gives

•               ∂T/∂z = (-A/k)z + c1                       (7.21)

                                                               39
     Calculation of Geotherms
• From the boundary condition ii, -k(∂T/∂z) = -Q0 at
  z = 0, (∂T/∂z) = Q0/k = c1 (Eqn 7.22).
• Substituting 7.22 into 7.21 and integrating again:

•    T = (-A/k)(z2/2) + (Q0/k)z +c2
                                           (7.23)
• From the boundary condition i, T = 0 when z = 0,
  c2 must be zero. The temperature in the column
  is then given by:

•          T = (-Az2/2k) + Q0z/k
                                             (7.24)
                                                  40
      Calculation of Geotherms
• An alternate set of boundary conditions could be used to
  estimate the equilibrium crustal geotherms given d as
  the depth of the moho and Qd as the mantle heat flow
  into the base of the crust:
   – T = 0 at z = 0
   – Q = -Qd at z = d
• The temperature in the column 0 ≤ z ≤ d is given by:

• T = (-Az2/2k) + [(Qd + Ad)/k]z
                                                     (7.28)
• Contributions to the surface heat flow come from both
  mantle (Qd) and internal crustal (Ad) sources; similarly,
  mantle heat flow contributes Qd/k to the temperature at
  depth z.
                                                              41
Calculation of Geotherms
            • Equilibrium geotherm: no
              change in temperature with
              time (steady-state); ∂T/∂t = 0
            • Depth 0 ≤ z ≤ d
            • T = 0 at z = 0
            • T = Td at z = d
            • No internal heat generation
              (A = 0)

            Eqn 7.20: ∂2T/∂z2 = -A/k = 0
            ∂T/∂z = (Td – 0)/(d – 0) = Td/d
            Integrate ∂T = (Td/d) ∂z  T = (Td/d)z



                                                 42
         Calculation of Geotherms:
            One Layer Models
                                  • d = 50 km, k = 2.5 W m-1 °C-1, Qd
                                    = 21 x 10-3 W m-2 *, A = 1.25 mW
                                    m-3, from Eqn 7.27.
                                  • a: standard model; shallow
                                       gradient ~30 °C km-1, deep
                                       gradient ≤15 °C km-1.
                                  • b: k reduced to 1.7 W m-1 °C-1;
                                       shallow gradient raised to
                                       ~45 °C km-1.
• c: A raised to 2.5 mW m-3; shallow gradient raised to >50 °C km-1.
• d: Qd doubled to 42 x 10-3 W m-2; shallow gradient raised to
     ~40 °C km-1.
• e: Qd halved to 10.5 x 10-3 W m-2; shallow gradient reduced to
     ~27 °C km-1.
                                    * page 281 erroneously reports this as
                                                                          43
                                    21 x 10-3 mW m-2
         Calculation of Geotherms:
            Two Layer Models
                                           • Eqns 7.20 – 7.28 apply
                                           • each layer considered
          z1
                                             separately
          z2
                                           • temperature gradients
                                             must be matched
                                             across boundaries



Boundary conditions:                      For the first layer 0 ≤ z < z1:
      i) T = 0 at z = 0                              ∂2T/∂z2 = -A1/k
      ii) A = A1 = 4.2 for 0 ≤ z < z1     For the second layer z1 ≤ z < z2:
      iii) A = A2 = 0.8 for z1 ≤ z < z2
      iv) Q = -Q2 = -63 at z = z2                    ∂2T/∂z2 = -A2/k
                                                                       44
            Calculation of Geotherms:
               Two Layer Models
                                             The solution to these two
                                             differential equations is subject
                                             to:
                                              Boundary conditions
                                              Matching both temperature
                                               and temperature gradient at
                                               z = z1
                                             is given as two equations:

   T = -A1z2/2k + {Q2/k + [A2/k(z2 – z1)] + A1z1/k}z    for 0 ≤ z < z1 (7.31)
  T = -A2z2/2k + {Q2/k + A2z2/k}z + [(A1 - A2)/2k]z12   for 0 ≤ z < z1 (7.32)
The calculated equilibrium geotherm from these equations for an Archaean
crust is shown above. This indicates rather high crustal temperatures
                                                                      45
existed during this time period (2.5 – 4.5 Ga) of the Earth.
Comparison
    of
Geotherms

The Archaean gradient
approaches the shape
of curve d, which has
double the heat flow Qd
than the standard
model, curve a,
indicating higher heat
flow in the Archaean      46
   Typical
   Craton
  geotherm

  Conductive to
  ~200 km
  (~6°C/km)




The geotherm beneath the Slave Province in Canada, as inferred from xenoliths from
the Jericho pipe. Squares are unsheared xenoliths, triangles are sheared xenoliths, and
asterisks are megacrystalline xenoliths. The conductive geotherm in the mantle
extrapolates to the base of the crust at X and to the mantle adiabat at L. The shallow
                                                                                    47
geotherm G is from borehole measurements. From Sleep 2005 AREPS
Timescale of Conductive Heat Flow
• Consider model rock column with gradient curve a from
  Fig. 7.3
• Increase Qd from 21 x 10-3 W m-2 to 42 x 10-3 W m-2;
  temperature of rock column would increase until new
  equilibrium temperatures were attained along curve d.
• This process occurs very slowly; consider a rock at z =
  20 km, with initial temperature T0 = 567 °C.
   – After 20 Ma conduction would have raised the temperature to T20
     = 580 °C.
   – After 100 Ma the temperature T100 > 700 °C.
   – The equilibrium temperature Te = 734 °C.


                                                                  48
Timescale of Conductive Heat Flow
• Theses values can be estimated using the
  thermal diffusion equation (7.17):
•             [∂T/∂t] = k2T
• The characteristic time t = l2/k indicate the
  amount of time it takes for a temperature change
  to propagate a distance of l in a medium having
  thermal diffusivity k.
• The characteristic thermal diffusion distance
  l = √(kt) estimates the distance in which
  changes in temperature propagate in a time t.
                                                49
Timescale of Conductive Heat Flow




• Example 1: thermal transfer from a subduction
  zone at z = 100 km via conduction >> 10s Ma
• Melting and intrusion are important mechanisms
  for heat transfer above subduction zones

                                               50
Timescale of Conductive Heat Flow




• Example 2: metamorphic belt caused by a deep heated
  heat source is characterized by abundant intrusions,
  often mantle-derived material.
• Magmatism occurs because large increases in deep
  heat flow cause large-scale melting at depth long before
  the heat can penetrate towards the surface by
  conduction.
                                                         51
               The Error Function
  • An integral that occurs often in the solution to
    physical problems:
                                   x
  •            erf(x) = (2/√p) ∫       e -y 2   dy    (7.35)
                                  0



                                                     Error function
                                                     table in
                                                     Appendix 5,
                                                     p.638-9




put equation
                                                                 52
on board
    Instantaneous Cooling or Heating
• A semi-infinite solid with upper surface
  z = 0 has no heat generation (A = 0) and
  an initial temperature throughout of T = T0
  at t = 0.
• Determine how the solid cools for t > 0
  given that for z = 0, T = 0 for t > 0.
• Use thermal diffusion equation (7.17):
•             [∂T/∂t] = k2T
                                            53
    Instantaneous Cooling or Heating
• Solution to the thermal diffusion equation that
  satisfies the boundary conditions:
•          T = T0 erf[z/(2√(kt))]            (7.34)
• The time taken to reach a given temperature is
  proportional to z2 and inversely proportional to
  k.
• The temperature gradient is given by
  differentiating (7.34) wrt z:

                                                     54
 Instantaneous Cooling or Heating

• ∂T/∂z = T0 ∙ (2/√p) ∙ 1/(2√(kt)) ∙ e-z/(4kt)
• ∂T/∂z = T0/(√(pkt)) e-z/(4kt)               (7.36)
• The error-function to the heat-conduction
  equation can be applied to many geological
  situations.
• Example: dike of 2w and infinite extent in y and z
  directions; no heat generation, ignore latent heat
  of solidification. Use: thermal diffusion equation.
                                                   55
 Instantaneous Cooling or Heating
• Initial conditions:
• T = T0 at t = 0 for -w ≤ x ≤ w;
• T = 0 at t = 0 for -w ≤ x ≤ w.
• Solution to the thermal diffusion equation that
  satisfies the initial conditions (Eqn 7.37):

• T = (T0/2){erf [(w – x)/(2√(kt))] +
                      erf [(w + x)/(2√(kt))]

                                                    56
 Instantaneous Cooling or Heating
• For w = 1m, intrusion temperature of T0 =
  1000 °C, and k = 10-6 m2 s-1, then the
  temperature at the center of the dike:
  – Tc = 640 °C after one week;
  – Tc = 340 °C after one month; and
  – Tc = 100 °C after one year.
• Note that small dikes cool rapidly


                                              57
 Instantaneous Cooling or Heating
• General case: temperature of the dike Tc
  – Tc = T0/2 when t = w2/k
  – Tc = T0/4 when t = 5w2/k
• High temperatures outside the dike are confined
  to a narrow zone:
  – Distance w from dike edge Tmax = ~T0/4
  – T0/2 reached about distance w/4 from dike edge
• Contact metamorphism generally confined to a
  narrow zone 2w (dike width) on either side.

                                                     58
        Dike Intrusion
    Contact Metamorphism




Golden Gate National Park,
Clarens, South Africa        59
    Dike Intrusion
Contact Metamorphism


                  T0 + 590 °C at ts




                                = ts




                                      60
 Instantaneous Cooling or Heating
• Example: Periodic Variation of Surface
  Temperature
• Earth’s surface temperature varies periodically (daily,
  annually, ice ages); we make temperature measure-
  ments deep enough to minimize these effects
• Periodic contribution to surface temp modeled as T0eiwt,
  where:
    T0 is the maximum variation of mean surface temperature;
    w is 2p multiplied by the frequency of variation; and
    i is (-1).


                                                                61
    Periodic Variation of Surface
            Temperature
• The temperature T(z, t) is given by the
  diffusion equation with the following
  boundary conditions:
  – T(0, t) = T0eiwt, and
  – T(z, t) → 0 as z → ∞
• The separation of variables technique is
  used is used to solve this problem, as
  explained in the text, and the solution for
  T(z, t) is equation 7.45 (p.284)
                                                62
     Periodic Variation of Surface
             Temperature
• Skin Depth L: the depth at which the amplitude
  of the periodic disturbance is 1/e of the
  amplitude at the surface. L is calculated by:
•              L = √(2k/(wrcP)                (7.46)
• Given a sandstone with k = 2.5 Wm-1C-1, cP =
  103 J kg-1 C-1, and r = 2.3 kgm-3, and the daily
  variation w = 7.27 x 10-7 s-1), L is ~17cm;
• For the annual variation w = 2 x 10-7 s-1,
  L = 3.3m
• For an ice age (with period of the order of
  100ka), L > 1km                                    63
    Periodic Variation of Surface
            Temperature
• Phase Difference f: difference between
  surface temperature variation and the
  variation at depth z is given by:
•         F = z√[(wrcP)/(2k)]          (7.47)
• At skin depth L, f = 1 radian
• When f = p, temperature at depth z is
  exactly half a cycle out-of-phase with the
  surface temperature.

                                               64
   Problem Set #5: Heat Flow
• Pages 321-323 and Additional below
• Problems: A-1, 5, 22, 2, 14 – 20: evens,
  A-2, A-3
• Show all work!
• Due 9 October 2009




                                             65

								
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