The Wine Cellar Problem - PowerPoint PPT Presentation

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The Wine Cellar Problem

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The Wine Cellar Problem Geophysics most important contribution to the human race. – PowerPoint PPT presentation

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Title: The Wine Cellar Problem


1
The Wine Cellar Problem
  • Geophysics most important contribution to the
    human race.

2
The Situation
x
?
z
3
Questions
  • What is the temperature anomaly as function of
    time, depth and the Fourier transform of qs(t)?
  • What constants determine the attenuation depth of
    the temperature anomaly?
  • What is the attenuation depth of the periodic
    temperature variations due to the
  • Diurnal cycle?
  • Annual cycle?
  • Glacial cycle?

4
Assumptions
  • The ground is a semi-infinite homogenous
    half-spaceso we use the 1-D, time dependent heat
    conducting equation
  • Constant thermal properties (k, k)
  • As z gt infinity the temperature T(z,t) gt To,
    where To is the average surface temperature
  • which means were ignoring heat flux from the
    mantle, and we have no internal heat sources
  • which essentially means the ground in question
    is an isolated body

5
Deriving the Temperature Anomaly
  • If qs(t) is a periodic forcing function we can
    assume it is of the form . So the
    differential equation at the surface becomes
  • Because the heat flux is periodic and the PDE is
    linear we can guess the solution has the form

6
Deriving the Temperature Anomaly
  • Substituting T(z,t) into the diffusion equation
    we get
  • Which reduces to a 2nd order linear ODE
  • Which has the well known general solution

7
Still Deriving
  • Because were interested in the exponential decay
    with increasing depth, we let a 0, then select
    the second term and plug f(z) back into T(z,t) to
    get

8
Still Deriving
  • And after separating out the oscillatory part

9
But what about A?
  • Apply the boundary condition at the surface
  • If we sub T(z,t) into this bad boy we get

10
And so our super final answer is
11
Finally, compare T(z,t) with q(s)
  • There is a difference of ?/4 between the
    oscillatory parts of these two functions

See the extra ?/4?
meaning that the temperature anomaly at any
given depth will lag behind the surface
fluctuation by 1/8 of the period of the
fluctuation.
12
Attenuation Depth
  • The depth at which the temperature has negligible
    fluctuation w.r.t. the surface temp. In other
    words where do we put our cellar??
  • where
  • Equate this to the temperature function

13
Attenuation Depth
  • and solve for z!
  • zo is only dependent on k and w
  • So re-write the temperature function

14
So now what?
  • We want to know how the attenuation depth will
    vary with time and soil conditions
  • so we chose three time scales to examine
  • w 2pf
  • Diurnal w 7.27x10-5 rad/sec
  • Annual w 1.99x10-7 rad/sec
  • Glacial w 1.99x10-12 rad/sec

15
  • and we chose three soil conditions to consider
    Clay Soil, Sandy Soil, Rock

Clay Soil Sandy Soil Rock
k (W/m2/k) 0.25 0.30 2.90
k (x10-6 m2/s) 0.18 0.24 1.43
16
Diurnal Cycle
  • Tiny attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 0.07 0.08 0.20
17
Diurnal Cycle
18
Glacial Cycle
  • Huge attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 425 491 1199
19
Annual Cycle
  • Practical attenuation depths!

Clay Soil Sandy Soil Rock
zo (meters) 1.35 1.55 3.79
20
Annual Cycle
  • We selected a wine-bearing region with
    substantial temperature fluctuations
    Canandaguia, New York
  • NEW YORK CITY?! Get a rope.
  • Annual DT 18 kelvin
  • Were assuming that the average surface
    temperature, To, is the optimum temperature for
    storing wine 55ºF.

21
Canandaguia Clay Soil
22
Canandaguia Sandy Soil
23
Canandaguia Rock
24
Cheers!
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