A: For an Economy, like say the U.S. where we have variables Employed Men, Price,Volume,Wages,Investment,Scale of Operation, Production Process Mean Dimensions of Radius R with Number of space dimensions being used = m, Dollars, A Gross National Product, Specific Products and the change in time of these variables then:

For a change in Production of a Product with time the Behavior of Men is to Minimize their time/Per unit of Volume of the product, i.e. that ratio (dMen/dRadius with time)=0.

Using reduced values, or indexed values of these variables based on a given year for GNP, Price, Volume, etc. and the Fact that Dollars=Price*Volume=Wages*Men, and Volume=Radius^m gives a Slope for the Experienc Curve=(1/m-1)*(1-Exp(-k*t)) wherE k=growth rate constant for the product and t=time (usually years) .

Because “All Men involved minimize their time per unit of space dimension, with time”.  Thus this Law of Nature is claimed as Garnett’s Minimization Law

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Inherent in the foregoing is also the fact that Investment=Scale^(1/m).

Once you accept that the investment for a tank or reactor or container scales up with the 1/3rd power of volume then the scale up with pressure naturally follows.

If the rate of reaction of a chemical reaction is proportional to pressure (P) then raising pressure will lower the volume required at the same time that it will raise the thickness of the walls to contain that higher pressure.  It follows that Investment will be:

Investment=k*(1/P)^(1/3)*P

Investment =k*P^(2/3)

If the reaction is proportional to the pressure squared (2 species->one species)

Investment=k*(P)^(1-2/3)=k*(P)^(1/3)

For a 3rd order pressure dependency on the reaction rate the Investment is independent of pressure, i.e.

Investment=k*P^(1-3/3)=k

For orders higher than 3 other things like heat transfer to/from the reactants (i.e. area or order 2),  or limiting conversion to limit temperature rise, or use of diluents to control temperature rise,  may become dominant, limiting the pressure effect between P^0 to P^(1/3).

By my memory the investment for polyethylene plants scale as the 1/4th power of Pressure.

The foregoing is applicable for a fixed production rate.  However if one is in the position of building a plant where the justification of the capital is a key element then one likes to consider investment per annual pound.  Larger plants tend to give lower investments/annual lb.  Since production rate is proportional to P raised to the power of 1,2, or 3 preceding the investment per annual lb will be proportional to Pressure raised to the power of -1/3, -5/6, -6/3=-2.  That being the case plants tend to be at the highest pressure and largest scale.

Why do scale up rules of thumb work and why is the historic six tenths power six tenths?

The cause and effect relationship is shown in this link (publication)

scale

Experience curves show the scale factor or number of dimensions “m” of the production facility, the growth rate constant constant, and contain cycles with time.  The full equation for the experience curves depends upon the scale of the production facility at a given time, the number of men currently working on it and the expansion effort currently underway.  Using men(t) as a variable, which is proportional to money via wages, and an exponential demand from the population gives the following differential equation and its’ solution.  The periodicity is shown to be a fundamental part of the solution.

the-full-experience-curve

I was talking to Gregory S. Patince recently about Research being an Investment, or not.  My view of course is that it can be if managed to be an investment.  Just how that is so is included in the following documents.  The criteria of what is required is defined.

research-as-investment-power-point1

research-as-investment-manuscript

These documents are part of the continuing dialog about what Experience Curves are and how they can be extended beyond historic data.

Will be at Houston Wed. to present to the American Institute of Chemical Engineering why Experience Curves have the Shape that they do and what the slope becomes for a mature product. It is because the slope of the x,y plot of log cumulative volume vs. log constant dollar price is (Scale factor – 1)*(1-exp(kg*time)).

The reason for the shape of experience curves is covered in this power point presentation. The material is more fully developed in the paper on the geometry of experience curves. The presentation went smoothly on Wed. 25, April at Houston. One question raised is “what happens to the experience curve for a product that is progressing nicely along an experience curve and the demand plateaus”. The answer is that it turns horizontal for a while since we are looking at constant dollar prices. The real prices rise along with inflation as time goes on. The mathematics shown for exponential growth of a product are applicable with the demand being specified as constant, i.e. the growth rate constant can be zero. For products in a ZPG (zero population growth) economy this situation will ultimately prevail.

In the presentation the question of “is learning the cause of the experience curve” was posed and answered in the negative; “it is not”. Both Professor Henderson and Michael Rothschild books do not prove that it is. An analysis of the only data basis in Rothschild’s Economy as EcoSystem can be shown to be best accounted for by Newton’s force equal mass*acceleration.