Exponential Curve 7

Exponential Curve 7

For the love of the human race.

Saturday, December 13, 2014

Our Thesis

We are greatly indebted to Dr. Albert Allen Bartlett (1923-2013), former emeritus professor of physics at the University of Colorado, Boulder.[1]  These are Dr. Bartlett’s ideas, we are merely reporting them.  We have performed a lengthy analysis of Dr. Bartlett’s “arithmetic” elsewhere.[2]  We now seek to summarize this “arithmetic” in one brief paper.

One cannot investigate either energy policy or energy theory without a thorough understanding of the exponential curve.[3]

Basic Exponential Equations

y = a * b^t

y = y₀ * b^t

P = P₀ * exp (-kt) = P₀ * e^-kt

Where, a, y₀, and P₀ are initial values; b = 1 + r; r is the decimal rate of increase or decrease over a set period of time (per year); t, is time; and, k is also an expression of increase or decrease.  For values less than 10%, r and k are nearly equal.

These curves approach practical infinity in brief periods of time.  Growth requires infinite energy.

Doubling Time

t (doubling time) ≡ ln (2) / ln (b)

t (doubling time) ≈ .7 / r (decimal) = 70 / r %

Doubling Amount

y = 1 * 2ⁿ

If accumulation is allowed the accumulation is one less than twice the most recent doubling: for example, if the most recent doubling results in a quantity of 32 units of anything, the accumulation is 64 – 1 or 63 units.  For all practical purposes the accumulation is twice as large as the most recent doubling.

y (accumulated) = 2 * 2ⁿ -1 ≈ 2 * 2ⁿ

Extraction Time

T_e (zero growth) ≡ A / y₀

T_e (growth) = 1 / ln (b) * ln [ ln (b) * A / y₀ + 1 ]

T_e (growth) ≈ 1 / r * ln [ r * A / y₀ + 1 ]

T_e (sustained availability) = ∞

T_e (practical sustained availability) ≈ around 5 half-lives

Extraction Peak

Peak (zero growth) ≡ y₀ (flat)

Peak (growth) ≡ y₀ * b^Te

Peak (sustained availability) ≡ y₀ (declining steadily)

Sustained Availability

k ≡ 1 / T_e

k ≡ 1 / A

t (half-life) ≡ ln(2) / k (rule of seventy)

Sustainability

0 ≤ CC+ HC₁ * P₁ + HC₂ * P₂ + HC₃ * P₃ + … + HC_n * P_n
≡ P_c1 * C_c1 + P_c2 * C_c2 + P_c3 * C_c3 + … + P_cn * C_cn ≤ Q

Conclusion

If you have followed the previous six discussions and know how to apply this simple “arithmetic”, this is all you need to verify any data or energy news you may run across.  When you find data, you will know what you are looking for.  All the “arithmetic” necessary to build a sustainable culture is right here.  The Gaussian family of curves are a little more difficult; you might need a mathematician to help you with these: but they are not necessary for checking for exaggerations, false information, incorrect arithmetic, or no arithmetic at all.

Fresh data will be reported separately so that it can be updated without revising every paper.  We are still in an energy crisis, and this is the method we will use to track and evaluate its progress.

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[1] http://en.wikipedia.org/wiki/Albert_Allen_Bartlett

[2] http://swantec-ep.blogspot.com/

[3] http://www.albartlett.org/presentations/arithmetic_population_energy_video1.html, and http://www.albartlett.org/presentations/arithmetic_population_energy_video2.html

[4] If you have been blessed or helped by any of these meditations, please repost, share, or use any of them as you wish.  No rights are reserved.  They are designed and intended for your free participation.  They were freely received, and are freely given.  No other permission is required for their use.