An Introduction to Heat Flow Lecture 10/15/2009 GE694 Earth Systems Seminar.
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Transcript of An Introduction to Heat Flow Lecture 10/15/2009 GE694 Earth Systems Seminar.
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.
Fourier’s Law of Heat Conduction:
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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 Fourier’s law. In order to apply Fourier’s law, the thermal conductivity constant k must be measured in the laboratory).
Heat Generation by Radioactive Decay:
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Distribution of Radioactive Elements in the Earth:
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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.
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.
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Note: T increases with y2 due to the internal heat production.
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)
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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.
Instantaneous Heating or Cooling of a Half-Space
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These are the initial conditions at t=0 and the boundary conditions at y=0 and y=infinity.
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Equation (4-94) is the 1-D diffusion equation rewritten in the new coordinate system.
Here, eta is called the “similarity variable”.
In terms of eta, the diffusion equation becomes
The solution to the differential equation (4-100) involves a special function called the “error function” or erf(x). The solution is
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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
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x
y
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).
Cooling of Melts: The Stefan Problem
The phase change releases heat, and so it acts as a heat source.
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= 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.
Cooling of Melts: The Solidification of a Dike or Sill
This problem starts with a temperature T=Tm in the dike and at its boundary, and the rest of the country rock is at T=T0. 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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