# Geothermics chap 17 LECTURE

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```					    Physics comparison: gravity and force and
acceleration, isostasy and pressure and buoyancy,
heat and temperature
•Gravity was application of a fundamental force law (of the 4 fundamental forces).

•The gravitational attractive force between masses causes motion that is
determined by Newton’s three laws of motion (Newtonian Mechanics).

•Isostasy uses the concept of a ideal-fluid in a gravitational field that produce the
force called ‘weight’. This leads to the concept of stress or more simply pressure.
This lead to the concept of a buoyancy force.

•Heat and temperature is a very different physics that applies Newtonian mechanics
to systems with 1024 molecules (atoms). A statistical approach is required as no one
can solve for the motion of 1024 molecules. Happily, we have discovered that in
thermal/chemical equilibrium thermodynamic averages of many-particle system is
well-defined so that macroscopic averages like temperature, heat content are
defined by simple equations.
Energy, Heat, Temperature, Work concept web
•Red=hot; blue=cold.

•Is this in thermal equilibrium ?

•How is equilibrum attained ?
First Law of Thermodynamics
The first law of thermodynamics is the application of the conservation of energy
principle to heat and thermodynamic processes:
Microscopic (Internal) Energy
Internal energy involves energy on the microscopic scale. For an ideal monoatomic
gas, this is just the translational kinetic energy of the linear motion of the "hard
sphere" type atoms , and the behavior of the system is well described by kinetic
theory. However, for polyatomic gases there is rotational and vibrational kinetic
energy as well. Then in liquids and solids there is potential energy associated with
the intermolecular attractive forces. A simplified visualization of the contributions to
internal energy can be helpful in understanding phase transitions and other
phenomena which involve internal energy.
Work (energy in transition)
•The amount of mechanical work done can be determined by an equation derived
from Newtonian mechanics: Work = Force x Distance moved in force direction

•It can also be described as the product of the applied pressure and the displaced
volume:       Work = Applied pressure x Displaced volume

•The unit of work is joule, J, which is defined as the amount of work done when a
force of 1 Newton acts for a distance of 1 m in the direction of the force.
1 J = 1 Nm

Solid earth, liquid oceans, and gaseous atmosphere are all heat engine that
do work on the planet (create kinetic energy) due to disequilibrium associated
with lateral variation in temperature.

•   Solid earth: hot bottom, cold top: convection moves heat to cool solid planet.
•   Liquid ocean: hot at equator (why?), cold at poles, pole ward flow moves heat
from tropics to polar regions.
•   Gaseous atmosphere: hot at equator , cold at poles, pole ward flow moves heat.
Mechanical Equivalent of Heat
Heat flow and work are both ways of transferring energy. As illustrated in the heat
and work example, the temperature of a gas can be raised either by heating it, by
doing work on it, or a combination of the two.

In a classic experiment in 1843, James Joule showed the energy equivalence of
heating and doing work by using the change in potential energy of falling masses
to stir an insulated container of water with paddles. Careful measurements
showed the increase in the temperature of the water to be proportional to the
mechanical energy used to stir the water.
Heat and Work Example

This example of the interchangeability of heat and work as agents for adding
energy to a system can help to dispel some misconceptions about heat. I found
the idea in a little article by Mark Zemansky entitled "The Use and Misuse of the
Word 'Heat' in Physics Teaching". One key idea from this example is that if you
are presented with a high temperature gas, you cannot tell whether it reached
that high temperature by being heated, or by having work done on it, or a
combination of the two.
Heat (e.g., steam) engines and Society
•Cellular metabolism: heat released in cells
•Electricity : heat engine, e.g., coal fired plant making steam to spin turbine
•Cars: heat production via compustion of air/gas mixture.
•House heating: burning natural gas or electricity
•Solid/liquid/gaseous earth envelopes are all heat engines.
•Energy as burning stuff to make heat to drive heat engines to extract motive force
Heat engines: extract motive power from temp. differences
Steam engines: temp. difference to motive power
Steam turbine energy (chemical/heat/mechanical/electrical): burn coal,
make heat to make steam that spins a turbine to spin an electrical
generator
Car engine
Newton’s cooling law

Q = Thermal energy (Joules)             h = heat transfer coeficient

A = area of the heat being transferred T = Temperature of the object

Tenv = Temperature of the environment

ΔT(t) = T(t) − Tenv : time-dependent thermal gradient between environment and object

Exponentially decay in time solution:

T(t) = TA + (TH-TA) e-kt
Fourier's
Heat
Conduction
Equation:
History,
Influence,
and
Connections
Inventing thermodynamics:
temperature and heat
What does heat flow equation conserve? Heat
Energy
Heat flow equation variations
Diffusion is a molecular scale phenomena everywhere
Solid state atomic diffusion   Thermal (heat) diffusion

1-dimensional random walk       2-D random walk
Diffusion equations

Chem concentration: Fick’s Law
Heat: Fourier’s Law
Current: Ohm’s Law
Porous flow: Darcy’s Law

Heat/concentration/charge or
Heat transfer requires ∆T ≠ 0
Conduction is diffusion of atomic vibrations at the atomic scale when
there is a thermal gradient. Conduction can never ever be stopped
except by the perfect insulator idealization.

Convection is the movement of heat by material flow. Convection
happens when the thermal variations creates buoyancy forces
sufficient to overcome the viscous strength of the mantle rock. The
earth is a very viscous fluid that flows at rates up to 10 cm/yr as
manifest by plate velocities.

Radiation is the movement of heat as infrared electro-magnetic waves
and hence can propagate through free-space and transparent
substances.

not EM-waves, but nuclear fragments. The heat comes from the huge
kinetic energy of the nuclear fragments being transfer into lattice
vibrations.
Heat and temperature
Discuss what you think the answer is to the following question: Which object
contains the most heat, a boiling pot of water or a gigantic iceberg?

Temperature is not energy, but a measure of energy intensity.
Q = C * T where Q is heat and C is specific heat material property.

Temperature is defined at a point (intensive property) and energy is defined
over a volume (extensive property).

Extensive: mass, volume, entropy, energy.
Intensive: temperature, chemical potential, density, pressure.
P * V = n *R *T
In the study of transport phenomena, flux is defined as the amount
that flows through a unit area per unit time[1] Flux is a vector.

Momentum flux, the rate of transfer of
momentum across a unit area (N·s·m−2·s−1).

Heat flux, the rate of heat flow across a unit
area (J·m−2·s−1).

Chemical flux, the rate of movement of
molecules across a unit area (mol·m−2·s−1).
(Fick's law of diffusion)

Energy flux, the rate of transfer of energy
through a unit area (J·m−2·s−1). The radiative
flux and heat flux are specific cases of energy
flux.
Convection in a box in a computer
Heat is kinetic/potential energy at atomic scale. The MKS unit for energy is Joules. A Joule has
units of a force times a distance (Newtons*meters) which is definition of work. Thus, energy is the
capacity to do work. Raising a 1 kg object 1 m in a 10 m/s2 gravitational acceleration field requires
that 10 Joules of work by performed on the object.

Temperature is NOT energy, but a measure of energy intensity. Temperature is mean kinetic and
potential energy of atoms/molecules. The MKS temperature unit is Kelvin (but Celsius and
Farenheight are related). Temperature is a measure of the tendency of an object to spontaneously
give up energy to its surroundings. When two objects are in contact, the one that tends to
spontaneously lose internal energy is at the higher temperature. When object have the same
temperature, they are in thermal equilibrium. At zero degrees Kelvin, all thermal motion ceases.

Simple relation between heat and energy (ignoring whether at constant pressure or temperature)
is Q (J) = C (J/K) * T (K) where Q is heat and C is heat capacity. To raise the temperature of 1
kilogram of water 1° Kelvin requires 4200 Joules of energy. Different substances have different
heat capacities proportional to their valence electron behavior and crystal lattice structure.

Which object has more heat energy: an iceberg at -2° C or a cup of boiling water at 100° C ?

While the iceberg is at lower temperature than the boiling water, the much larger volume of the
iceberg with respect to the boiling cup of water, means it has more total heat energy (Joules).

Q = C * T where C is proportional to the volume of the object.
If a temperature contrast exists, heat will be transferred from the high to low
temperature by conduction and radiation and maybe convection if the force/strength
ratio in the mantle is sufficient to drive heat advection faster than diffusion.

This inexorable process of heat transfer only ends when the temperature contrasts is
no more (∆T=0° ), which defines thermal equilibrium

WHY MUST HEAT FLOW? All forms of energy (e.g., heat), are relentlessly driven to
spread out (disperse) their energy into larger volumes over time. This energy
spreading out drives a system toward thermal equilibrium. This truism is called the
second law of thermodynamics which says entropy always increase in time.
Terrestrial heat power:
44 TW
Flux: 0.04 W/m2           Heat flow:

Solar mean heat power:    Red (high 200 mW)
Blue (low 20 mW)
174 pW or 174,000 TW
Flux: 680 W/m2

Solar input 20,000
times more than 44 TW
Heat Flux in milli-Watts (mW)
Temperature at 3 km depth

This map is made by solving the heat flow equation for dT(z)/dz:

q (W/m2 ) = K * dT(z)/dz
Geologic phenomena and heat flow
•Yellowstone hydrothermal geyser fields
•Oil/gas maturation
•Volcanoes
•Earthquakes
•Hydrothermal mineralization
•Plate tectonics
•Geothermal power production
•Mid-ocean ridges
•Subducting slabs
•Metamorphism of rocks
Chapter 17 Geothermics
• The earth is hot at the core-mantle boundary ( about 4000° C) and cold at its surface (19 °
C) and very cold in space (-270 ° C). This large temperature contrast means the earth is in
thermal dis-equilibrium and must be transferring heat into space.

• The internal earth is cooled by convection that delivers heat to the base of the
lithosphere where the heat is then conducted through the lithosphere. This heat flux
warms the atmosphere and finally the atmosphere is always radiating heat into space.

• Note: the Earth primarily cools itself by creating new lithosphere at ridges that is pulled
across the ocean basin and finally subducted deep into the mantle to cool it.

• The solid earth is a heat engine because work is done moving the plates and raising the
mountains and pulling apart the ridges/rifts. The ocean-atmosphere is also a heat engine.

• In the last 4 Ga, the earth’s cooling has been about 100 ° C per Billion years.

• The earth is predicted to reach thermal equilibrium with space in about 10 Ga.
Three heat transport processes
There are three ways to transfer heat

Conduction: heat in a solid is transferred via
diffusion of crystal lattice vibrations. Conduction
can never be stopped if a temperature difference
exists between two regions. Examples?

Convection: heat is moved (advected) by different
density parcels flowing up and down in the
earth’s gravity field. This only occurs if a parcels
buoyancy force (Archimedes principle) is
sufficiently large with respect to the strength
(viscosity) of the mantle. Examples?

electromagnetic waves at speed of light. All
matter at non-zero temperatures radiates energy.
Examples?
Radiative heat flux from Planet earth
Energy (N*m or J) Power=energy/time (J/s or W) Flux = Power/area (J/s-m2 or W/m2 )
Convection simulation of earth
Heat at the atomic scale: lattice vibrations in solids
Conductive heat transport in lithosphere
and convective heat transport in mantle and fluid outer core
The lithosphere is strong and hence translates as a quasi-
rigid block over the asthenosphere, therefore heat cannot
be convected across the lithosphere and instead heat is
conducted across the lithosphere.

The mantle dominantly moving heat via convection,
although heat conduction is always occurring. But the 1-10
cm/yr mantle flow rates moves (advects) the heat much
faster than heat conduction!

The temperature with respect to depth (geotherm) in the
lithosphere is near a straight line which is called the
conductive geotherm: dT(z)/dz = 30°C/km. The
lithospheric geotherm will have curvature when there is
significant heat production (e.g., in continental crust).

temperature rise with depth is because each parcel of heat
is being compressed into a smaller volume due to the
pressure increase with depth.
•adiabatic - a process where a parcels temperature
changes due to an expansion or compression, but no heat
is added or taken away from the parcel.

Mantle Potential temperature
Heat flux and amount of heat power
Little-q Heat flux: W/m2
 W   K  W  * T  C 
q 2 
Power (W) per area (m2 )                                     
Energy (J) per time (s) per area (m2)    m        mC  z  m 
 W  * T  C  * A m 2
   
Big-Q Heat Power: W
Energy (J) per time (s)                 Q W   K      
 mC  z  m 
Geotherm with 0° C at surface (z=0 m)
Heat flux through area A
dT

dz                      Conductive
lithosphere

∆z =
Convective
mantle
35° C per km
Measuring thermal conductivity
Because it is very hard to measure mW scale
heat fluxes at earth’s surfacee, a tactic is to
Backwards!   measure the temperature gradient in a
q                       borehole and the thermal conductivity of the
rocks in the borehole.

Heat flux is calculated: q = k*dT/dz (W/m2 )

The thermal conductivity of a rock can be
measured by heating one side of the metal rod
and cooling the other side and measuring the
thermal gradients in the metal and rock-sample
regions.

Using the known conductivity of the metal, the
heat flux is calculated.

Thus, the conductivity of the rock is:
K = q / (dT/dz).
Thermal conductivity   Thermal conductivity
Material
(cal/sec)/(cm2 C/cm)   (W/m K)*
Thermal conductivity              Diamond            ...                    1000
Silver             1.01                   406.0
values                      Copper             0.99                   385.0
Gold               ...                    314
Good heat conductors:             Brass              ...                    109.0
Diamond, Metals                   Aluminum           0.50                   205.0
Iron               0.163                  79.5
Steel              ...                    50.2
Poor heat conductors              Lead               0.083                  34.7
Wood, Wool, Fiberglass            Mercury            ...                    8.3
Ice                0.005                  1.6
Conductivity is largely           Glass,ordinary     0.0025                 0.8
Concrete           0.002                  0.8
determined by ability of
Water at 20° C     0.0014                 0.6
electrons to wandered through a   Asbestos           0.0004                 0.08
substance. Good heat              Snow (dry)         0.00026                ...
conductors are also good          Fiberglass         0.00015                0.04
electrical conductors.            Brick,insulating   ...                    0.15
Brick, red         ...                    0.6
Cork board         0.00011                0.04
Wool felt          0.0001                 0.04
Rock wool          ...                    0.04
Polystyrene
...                    0.033
(styrofoam)
Polyurethane       ...                    0.02
Wood               0.0001                 0.12-0.04
Heat flow as function of oceanic plate age

The plate moves horizontally away from the mid-ocean ridge and cools as heat is
conducted into the ocean bottom over 150 ma.

This cooling is manifest as the cool lithosphere getting thicker with time.
Heat production: heat flux increases in direction of heat flow
Note that the heat flux vectors becomes longer
(bigger) towards the cooler side of the column.

This is because a constant heat production
(μW/m3 ) throughout the column is assumed.

This creation of heat increase the heat flow and
the thermal gradient towards the cooler end (i.e.,
the earth’s surface).

Thus, the distribution of heat production in the
crust changes the geotherm to make it a curve
and not a straight line for a steady state thermal
equilibrium.
Crustal heat productivity vs. heat flux relationship

The continental crust is highly concentrated in radioactive
elements such as Uranium. As Uranium decays, this is
termed heat productivity in micro-Watts per unit volume.

By measuring the surface rock heat productivity, a linear
relation between heat productivity and heat flux is often
found. This means crustal heat production controls the heat
flux.

The y-intercept value is the reduced heat flow (qr ) which is
heat flux from mantle convection.
Oceanic and continental heat flux and geotherms
The geotherm in the
lithosphere is conductive and
the geotherm in the convecting

Note that the conductive heat
flux with respect to depth in
the oceanic lithosphere is
constant and hence the
geotherm is a straight line.

But, the conductive heat flux in
the continental lithosphere
increases towards the surface
and the geotherm is a curved
line.

The base of the lithosphere has
the same temperature at all

Note that 40% of the Earth’s total heat flow

The other 60% of heat comes from ‘secular
cooling’ of the original accretional heat
when the Earth formed 4.5 Ga ago.

Also, note that 10% of the heat comes from
continental crust, but only 0.15% from
ocean crust.

In general, there is a decrease in heat flow
with the age of continents. Why ?
Planetary heat budget
46 +/- 3 TW (Terra-Watts)

Terra is 1012

World power consumption is 15 TW.

The primary contributions to
observed total surface heat are
shown.

mantle cooling

heat flow from the core

These three heat sources dominate
the mantle energy budget, but there
are substantial uncertainties in the
latter two contributions.
Mantle temperature and melting

By knowing the approximate temperature at which mantle rocks melt, the geotherm
can be compared to the mantle solidus to see if melt should form.
Seismic velocity mapped to temperature for Oceanic
lithosphere
Temperature re-equilibration

So far, we have been assuming that the
temperature does NOT vary in time. This
is called thermal steady-state or dynamic
equilibrium.

So, what if the temperature structure
changes in time due to basins filling or
tectonic overthrusts?

Adding sediments quickly to the top of
the crust, causes a thermal disequilibrium
that will be brought to thermal
equilibrium over Ma time-scales.

Thrusting 20 km of the upper crust over
the surrounding surface quickly, creates a
disequilibrium that takes greater than 50
Ma to reach thermal equilibrium.
Geothermal heat extraction and energy
production
Where high temperature rock is near the
surface, the heat contained in the hot-rock
can be mined to drive steam-turbines to
make electricity.

This is done by pumping water down the
injection well where it flows through the
cracks in the rocks and heats up and
corresponding cools down the rocks over
time. The hot water is then extracted via a
production well.

Note: geothermal energy is technically not a
renewable resource as the water circulation
extracts the heat much faster than the
earth’s mW scale heat flux can replace the
mined heat (energy).
Atmospheric temperature changes effects on shallow
geotherms

Periodic (daily, seasonal) changes in atmospheric temperature makes ‘temperature waves’
that propagate downward into the crust, but the attenuate exponentially with depth.

A recent 2 ° C increase in surface temperature between 1900-2000 can be measured in
the borehole temperature profile and well modeled using the heat flow equations.
END chapter
Thermal boundary layer and at core-mantle boundary
and heat exchange
Plume vs. Plate convection models
Starting subduction convection

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