An Introduction to Heat Flow

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An Introduction to Heat Flow

• Lecture 10/15/2009
• GE694 Earth Systems Seminar

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Basic Concepts

• Heat is a form of energy, and so the basic
  equations that describe heat and heat flow come
  from the conservation of energy law of physics.
• Heat Transfer Heat can be transferred by thermal
  conduction, thermal convection and radiation. In
  the solid Earth, the most important form of heat
  transfer is thermal conduction.

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Fouriers Law of Heat Conduction
The above solution is for a linear change of
temperature with distance. The change of T with
y can be nonlinear, which means that q can vary
with y.
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• The thermal conductivity of rocks is relative
  low, and is fairly similar for many different
  rock types

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North America Surface Heat Flow
Average continental heat flow 65 1.6
mW/m2 Average oceanic heat flow 101 2.2
mW/m2 Total continental heat flow 1.3 x 1013
W Total oceanic heat flow 3.13 x 1013 W (Heat
flow values are measured by drilling into rock,
measuring the temperature at different depths,
and then calculating q from Fouriers law. In
order to apply Fouriers law, the thermal
conductivity constant k must be measured in the
laboratory).
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Heat Generation by Radioactive Decay
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Distribution of Radioactive Elements in the Earth
In the crust and mantle, radioactive elements are
not distributed uniformly but rather concentrate
in some continental rocks. The radioactive
elements in crustal rocks also show a marked
difference when compared with the amounts of
these elements in chondritic meteorites.
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Steady-State Heat Flow when There is Heat
Production
flow of heat out of slab flow of heat into slab
heat production in slab (conservation of energy)
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The above differential equation can be solved by
integration with respect to y.
Note T increases with y2 due to the internal
heat production.
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Example Continental Geotherms
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3-D Steady-State Heat Flow with Heat Production
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1-D Time Dependent Heat Flow (no Heat Production)
Equations (4-67) and (4-68) are 1-D forms of the
diffusion equation, which shows up in many
different kinds of problems in physics,
chemistry, geophysics, geology, etc.
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Instantaneous Heating or Cooling of a Half-Space
These are the initial conditions at t0 and the
boundary conditions at y0 and yinfinity.
Equation (4-94) is the 1-D diffusion equation
rewritten in the new coordinate system.
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The solution to the differential equation (4-100)
involves a special function called the error
function or erf(x). The solution is
Here, eta is called the similarity
variable. In terms of eta, the diffusion
equation becomes
The boundary conditions become
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Cooling of the Oceanic Lithosphere A Half-Space
Model
Equation (4-125) can be used to estimate the
temperature T at depth y as a function of x (or t
since steady plate spreading is assumed). The
thickeness of the thermal boundary layer yL is
x
y
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Cooling of the Oceanic Lithosphere A Plate Model
The boundary conditions for a plate heated from
below, where the plate thickness at large time is
yL0, is
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Thus, at small times (i.e., near the spreading
ridge), this solution can be manipulated into the
half-space cooling model solution, while at large
times the solution becomes a simple linear
temperature gradient between the surface and the
bottom of the plate. For this latter case, the
heat flow is a simple conduction solution (4-134).
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Cooling of Melts The Stefan Problem
The phase change releases heat, and so it acts as
a heat source.
ym
?
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Solutions for this problem are found by matching
the values of the left and right hand sides of
equation 4-141 by trial and error.
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Cooling of Melts The Solidification of a Dike or
Sill
This problem starts with a temperature TTm in
the dike and at its boundary, and the rest of the
country rock is at TT0. As time increases, the
temperature of the dike cools and the heat
diffuses into the country rock, as in our earlier
problem of a sudden temperature increase at the
edge of a half-space. Skipping the steps of the
derivation to the solutions, we get a
transcendental equation to be solved by trial and
error.
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